广义case-cohort设计下多类型事件数据的加乘风险回归模型及其应用

广义case-cohort设计下多类型事件数据的加乘风险回归模型及其应用 Additive-Multiplicative Hazards Model for Generalized Case-Cohort Designs with Multiple Type Event Data and Its Application 1 2 first-author 刘君娥，女，博士，副教授，现从事数理统计及其应用方面的研究。 E-mail： xun4025@126.com 刘君娥，女，博士，副教授，现从事数理统计及其应用方面的研究。 E-mail： xun4025@126.com xun4025@126.com 3 淮北师范大学 经济与管理学院， 安徽 淮北 235000 淮北师范大学 经济与管理学院， 安徽 淮北 235000 School of Economics and Management， Huaibei Normal University， Huaibei 235000， Anhui， China School of Economics and Management， Huaibei Normal University， Huaibei 235000， Anhui， China 安徽省高校管理大数据研究中心， 安徽 淮北 235000 安徽省高校管理大数据研究中心， 安徽 淮北 235000 Anhui Big-Data Research Center on University Management， Huaibei 235000， Anhui， China Anhui Big-Data Research Center on University Management， Huaibei 235000， Anhui， China 集瑞联合重工有限公司， 安徽 芜湖 241009 集瑞联合重工有限公司， 安徽 芜湖 241009 C&C Trucks Co. Ltd.， Wuhu 241009， Anhui， China C&C Trucks Co. Ltd.， Wuhu 241009， Anhui， China 针对广义case-cohort设计下多类型事件数据的边际加乘风险模型，利用逆概率加权的思想建立加权估计方程，证明参数估计的相合性和大样本性质。数值模拟试验结果表明广义case-cohort设计在疾病事件发生率较高时更加具有应用性。 实例分析展示了其理论意义和应用价值。 Using the idea of inverse probability weighting, we first give a weighted estimating equation for an additive-multiplicative hazards model under generalized case-cohort designs with multiple type event data in this paper. The resulting estimators are shown to be consistent and asymptotically normal. Some numerical simulation experiments show that the generalized case-cohort designs is applicable when the incidence of disease is high. Finally, a real example is illustrated to show its theoretical significance and application value. 多类型事件数据 case-cohort设计 估计方程 multiple type event data case-cohort design estimating equation 0　 引 言 1 多元失效时间数据经常出现在生物、医学、社会学和经济学等研究领域中。例如，在流行病学队列研究中，以家庭为单位记录疾病发生的时间；临床医学中，单个个体可能经历几种不同类型的疾病；可靠性测试中，一台机器可能发生多次故障等。但是要详细地收集全队列成员的协变量信息，耗费成本较大。而case-cohort设计 ［ 1 1 ］ 只需要收集子列和子列外所有发病者的协变量信息，不需要收集子列外未发病个体的信息。为了节省资源，因此许多研究者将case-cohort设计引入到生物医疗研究中，从而产生了case-cohort设计下的多元失效时间数据。 对于case-cohort设计下的多元失效时间数据的研究，目前的成果大多是基于边际模型方法建模。Lu 等 ［ 2 2 ］ 对case-cohort设计下的成组失效时间数据，考虑了一类边际Cox比例风险模型，但在建模的过程中未考虑子列外失效个体的协变量信息。为了更充分地利用协变量信息，Zhang等 ［ 3 3 ］ 在边际Cox模型的基础上，建立了一类更有效的估计方程，获得了估计参数，改进了模型的有效性。Kang等 ［ 4 4 ， 5 5 ］ 和Kim等 ［ 6 6 ］ 分别用边际Cox风险模型和边际加性风险模型分析了case-cohort设计下多类型事件数据。还有部分学者基于加乘风险模型研究case-cohort数据，如Sun等 ［ 7 7 ］ 研究了case-cohort设计下带有时间相依协变量的加乘风险模型；周洁 ［ 8 8 ］ 针对加乘风险模型提出了一类带有时间相依权重的双重加权估计；Liu等 ［ 9 9 ］ 对case-cohort设计下多类型事件数据的加乘风险模型提出了一类估计方程；为充分利用子列外发生其他类型失效个体的协变量信息，刘君娥等 ［ 10 10 ］ 提出了一类有效的加权估计方程等。 然而，当现实中疾病的发病率较高时，子列外发病个体较多。针对这种情形，Chen ［ 11 11 ］ 提出了广义case-cohort设计。广义case-cohort设计是在一般case-cohort设计的基础上对子列外的发病个体再次进行抽样，这样只需抽取子列和子列外部分失效个体信息。目前对广义case-cohort设计下的多元失效时间数据的研究成果有限。Zheng等 ［ 12 12 ］ 基于广义加乘危险率模型，用拟得分方程方法对于多元数据的病例队列设计进行参数估计，并给出所得估计量的渐近性质。徐达等 ［ 13 13 ］ 利用加权估计方程方法，在广义病例-队列设计方案下，用Cox模型对长度偏差数据进行了分析和研究。基于此，本文将多类型疾病失效时间数据的加乘风险模型推广到广义case-cohort设计下。首先建立加权估计方程，然后给出参数估计的相合性和大样本性质，最后数值模拟试验结果表明广义case-cohort设计在疾病事件发生率较高时更加具有应用性。 1　 模型与估计方程 1 假设在某个队列研究中，有 n 个独立的个体，而对每个个体我们感兴趣的有 K 种类型疾病。设 <math id="M1"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929651&type= 4.23333359 3.72533321 ， <math id="M2"><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929444&type= 4.23333359 3.72533321 和 <math id="M3"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929462&type= 4.23333359 3.72533321 分别表示第 i 个个体第 k 类型疾病的潜在发病时间、删失时间和观察时间，并且 <math id="M4"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929478&type= 28.10933495 4.48733330 。 <math id="M5"><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929499&type= 27.26266861 4.57200003 是示性函数， <math id="M6"><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929515&type= 4.14866638 3.64066648 =1时表示失效，否则为删失。 令 <math id="M7"><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>≤</mo><mi>t</mi><mo>,</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929533&type= 42.58732986 4.57200003 表示第 i 个个体第 k 类型事件计数过程， <math id="M8"><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>≥</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929550&type= 30.14133072 4.57200003 表示风险过程， <math id="M9"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929570&type= 10.41399956 4.48733330 和 <math id="M10"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929590&type= 9.05933285 4.48733330 分别为 p 维和 q 维的协变量向量， <math id="M11"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929651&type= 4.23333359 3.72533321 和 <math id="M12"><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929618&type= 4.23333359 3.72533321 关于 <math id="M13"><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929638&type= 28.61733437 4.65666676 条件独立。 假设 <math id="M14"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929651&type= 4.23333359 3.72533321 的边际风险函数 ［ 14 14 ］ 为 <math display="block" id="M15"><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929665&type= 89.74666595 4.91066647 其中 <math id="M16"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929683&type= 22.18266678 4.91066647 是未知的 p + q 维的回归参数， g 和 h 是已知的连接函数， <math id="M17"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929705&type= 34.62866592 4.48733330 是未知的基本风险函数。 显然，当 <math id="M18"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>x</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929721&type= 15.32466698 3.80999994 ， <math id="M19"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929736&type= 14.47799969 3.72533321 时，模型（1）即为Cox比例风险回归模型。当 <math id="M20"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929756&type= 14.64733315 3.72533321 ， <math id="M21"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929766&type= 14.30866718 3.55599999 时，模型（1）即为加性风险回归模型。 令 <math id="M22"><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929781&type= 112.60665894 6.94266701 是关于事件历史信息 <math id="M23"><msub><mrow><mi mathvariant="normal">ℱ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>σ</mi><mo stretchy="false">{</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>:</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>u</mi><mo>≤</mo><mi>t</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929794&type= 85.17465973 4.65666676 的鞅，并且 <math id="M24"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930881&type= 1.60866666 2.79399991 表示调查终止时间。 假设从全队列中随机抽取 <math id="M25"><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929844&type= 1.86266661 2.79399991 个个体作为子列，令 <math id="M26"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930079&type= 9.31333351 4.57200003 表示第 i 个个体被选入子列，否则 <math id="M27"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930112&type= 9.31333351 4.57200003 。令 <math id="M28"><msub><mrow><mi>π</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>/</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929895&type= 39.79333496 4.82600021 表示第 i 个个体被选入子列的选择概率。 广义case-cohort设计在随机抽取子列后，接下来对子列外的所有发病个体进行抽样。对于第 k 类型疾病，通过简单随机抽样选取 <math id="M29"><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929921&type= 5.58799982 3.04800010 个个体。设 <math id="M30"><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929974&type= 3.47133350 4.57200003 是示性函数， <math id="M31"><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929974&type= 3.47133350 4.57200003 =1时，表示子列外发生第 k 类型疾病的第 i 个个体被选入子列，否则 <math id="M32"><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929974&type= 3.47133350 4.57200003 =0。设 <math id="M33"><msup><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929983&type= 4.57200003 3.04800010 和 <math id="M34"><msup><mrow><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20929994&type= 4.57200003 3.04800010 分别表示全队列和子列中发生 k 类型疾病的总人数，则 <math id="M35"><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>m</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>/</mo><mo stretchy="false">(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><msup><mrow><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930008&type= 76.19999695 5.07999992 表示子列外发生第 k 类型疾病的第 i 个个体被选入子列的选择概率。注意，（ <math id="M36"><msub><mrow><mi>η</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>n</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930022&type= 21.16666603 4.57200003 ）之间是相关的，而对于 <math id="M37"><mi>k</mi><mo>≠</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930034&type= 8.72066593 3.30200005 ，（ <math id="M38"><msub><mrow><mi>η</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo>…</mo><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>n</mi><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930047&type= 19.21933365 4.57200003 ）与（ <math id="M39"><msub><mrow><mi>η</mi></mrow><mrow><mn mathvariant="normal">1</mn><msup><mrow><mi>k</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></msub><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>k</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>n</mi><msup><mrow><mi>k</mi></mrow><mrow><mi>'</mi></mrow></msup></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930062&type= 22.35199928 3.80999994 ）是相互独立的。在广义case-cohort设计下，协变量的信息仅仅从子列和子列外抽取的发病个体上获得。因此，当 <math id="M40"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930079&type= 9.31333351 4.57200003 或 <math id="M41"><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930090&type= 10.32933331 4.57200003 时，观测到的数据为 <math id="M42"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>,</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930100&type= 38.26933289 4.65666676 ；当 <math id="M43"><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930112&type= 9.31333351 4.57200003 和 <math id="M44"><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930124&type= 10.32933331 4.57200003 时，观测到的数据为 <math id="M45"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930141&type= 13.63133430 4.48733330 。 为了建立广义case-cohort数据的估计方程，本文将使用下面的权重函数 ［ 4 4 ］ <math display="block" id="M46"><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msubsup><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930157&type= 81.61866760 4.91066647 其中， <math id="M47"><msub><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930174&type= 80.34867096 8.38199997 表示对第 k 类型事件，在 t 时刻子列中在险个体数占全队列在险个体数的比例； <math id="M48"><msub><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930189&type= 88.64600372 8.38199997 表示 t 时刻子列外抽取的第 k 类型事件失效个体数占子列外全体第 k 类型事件失效个体数的比例。 权重函数（2）实际上是利用逆选择概率进行加权，于是我们建立如下加权估计方程 <math display="block" id="M49"><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930203&type= 125.81465912 8.38199997 其中 <math id="M50"><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930214&type= 121.66600037 8.38199997 。并且对任意 <math id="M51"><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930224&type= 7.95866632 3.30200005 ，都有 <math id="M52"><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow></munder><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow></munder><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930239&type= 42.84133148 6.77333355 。 回归参数 <math id="M53"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930612&type= 3.04800010 3.72533321 的估计定义为方程 <math id="M54"><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930255&type= 15.32466698 3.55599999 的解，我们用 <math id="M55"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930262&type= 1.94733346 3.64066648 来表示。令 <math id="M56"><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930272&type= 32.59666443 6.94266701 ，则累积基本风险函数的估计量为 <math display="block" id="M57"><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">{</mo><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">β</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mi mathvariant="normal">d</mi><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mover accent="true"><mi mathvariant="bold-italic">γ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930278&type= 98.04399872 17.94933319 2　 渐近性质 1 为给出参数的渐近性质，假设以下正则条件成立： （C1） <math id="M58"><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>:</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930284&type= 55.45666885 4.48733330 独立同分布，其中 <math id="M59"><msub><mrow><mi mathvariant="bold-italic">T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mi>K</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930290&type= 35.47533035 4.65666676 ， <math id="M60"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>i</mi><mi>K</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930295&type= 35.47533035 4.65666676 ， <math id="M61"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>i</mi><mi>K</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930305&type= 49.36066818 4.65666676 ，并且 <math id="M62"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mi>K</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930311&type= 54.69466782 4.65666676 。 （C2） 对 <math id="M63"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930756&type= 22.43666649 3.21733332 ，都有 <math id="M64"><mi>P</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930334&type= 30.31066895 4.48733330 ，并且 <math id="M65"><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930346&type= 18.37266731 4.48733330 。 （C3） <math id="M66"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930356&type= 8.55133343 4.48733330 ， <math id="M67"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930366&type= 9.90600014 4.48733330 的任何一个组合以及 <math id="M68"><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930377&type= 14.39333248 4.48733330 都是有界的，并且几乎处处是 <math id="M69"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931068&type= 8.97466660 3.55599999 上的有界变差函数。 （C4） g 和 h 是连续可微的，并且对 <math id="M70"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930401&type= 45.21200180 3.21733332 ， <math id="M71"><mo>∂</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>,</mo><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930408&type= 38.35400009 4.82600021 在 <math id="M72"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930421&type= 2.96333337 3.72533321 的某个邻域内是同等连续的。 （C5） 设矩阵 <math display="block" id="M73"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mi>E</mi><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr><mtr><mtd><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930434&type= 108.96600342 11.17599964 则矩阵 <math id="M74"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931312&type= 2.79399991 2.79399991 是非奇异的，并且 <math id="M75"><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930461&type= 11.93799973 4.48733330 是 <math id="M76"><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930476&type= 14.22399998 4.40266657 的极限。 （C6） 令 <math id="M77"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930485&type= 14.73200035 4.65666676 ，其中 <math id="M78"><mi>α</mi><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930497&type= 14.30866718 3.55599999 。 （C7） 对所有 <math id="M79"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930506&type= 18.71133423 3.21733332 ，都有 <math id="M80"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930516&type= 16.84866714 4.65666676 ， <math id="M81"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930531&type= 19.64266586 7.87400007 ，其中 <math id="M82"><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930550&type= 15.49400043 4.65666676 ， <math id="M83"><msub><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930566&type= 15.74800014 4.65666676 。 在上述正则条件下，我们有下列关于 <math id="M84"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930584&type= 1.94733346 3.64066648 的渐近性质。 定理1 在条件（C1）~（C7）下， <math id="M85"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930599&type= 1.94733346 3.64066648 是 <math id="M86"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930612&type= 3.04800010 3.72533321 的相合估计。此外， <math id="M87"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930626&type= 18.11866570 5.41866684 依分布收敛于均值为0的正态分布，其协方差矩阵为 <math display="block" id="M88"><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow><mrow><mi>α</mi></mrow></mfrac><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mi>P</mi></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930638&type= 115.73933411 8.46666718 其中矩阵 <math id="M89"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931312&type= 2.79399991 2.79399991 如（C5）中所定义，并且 <math display="block" id="M90"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930661&type= 81.11067200 8.38199997 <math display="block" id="M91"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930680&type= 118.11000061 8.72066593 <math display="block" id="M92"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">(</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930693&type= 155.10932922 8.72066593 <math display="block" id="M93"><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930704&type= 119.63400269 4.91066647 <math display="block" id="M94"><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930726&type= 31.24200058 8.72066593 进一步提出 <math id="M95"><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930736&type= 9.56733322 5.16466665 （ <math id="M96"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930756&type= 22.43666649 3.21733332 ）的渐近性质。 定理2 在条件（C1）~（C7）下，对所有 <math id="M97"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930756&type= 22.43666649 3.21733332 ， <math id="M98"><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930768&type= 9.56733322 5.16466665 依概率收敛到 <math id="M99"><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930780&type= 9.56733322 4.48733330 ，对 <math id="M100"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931698&type= 13.97000027 3.55599999 一致地成立。此外， <math id="M101"><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>K</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>K</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930812&type= 115.56999969 5.41866684 弱收敛到一个零均值的高斯过程 <math id="M102"><mi mathvariant="bold-italic">𝒲</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930826&type= 9.39799976 3.55599999 。 下面将给出定理1和定理2的证明。为证明所提出估计的渐近性质，首先给出两个常用的引理 ［ 5 5 ］ 。 引理1 设 <math id="M103"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930927&type= 8.72066593 4.48733330 和 <math id="M104"><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930861&type= 9.05933285 4.48733330 是两个有界过程序列，对某个常数 <math id="M105"><mi>τ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930881&type= 1.60866666 2.79399991 ，使得 （a） 对某个有界过程 <math id="M106"><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930896&type= 7.45066643 3.55599999 ，有 <math id="M107"><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mn mathvariant="normal">0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>τ</mi></mrow></munder><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mover accent="false"><mo>→</mo><mrow><mi>p</mi></mrow></mover><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930911&type= 43.01066971 6.60400009 ， （b） <math id="M108"><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930927&type= 8.72066593 4.48733330 在 <math id="M109"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931068&type= 8.97466660 3.55599999 上单调， （c） <math id="M110"><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930959&type= 9.05933285 4.48733330 收敛到均值为零的过程，并且该过程有连续的样本轨道，则 <math display="block" id="M111"><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mn mathvariant="normal">0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>τ</mi></mrow></munder><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo></mrow><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mover accent="false"><mo>→</mo><mrow><mi>p</mi></mrow></mover><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930976&type= 61.72200012 7.61999989 <math display="block" id="M112"><munder><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">p</mi></mrow><mrow><mn mathvariant="normal">0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>τ</mi></mrow></munder><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mo stretchy="false">(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover accent="false"><mo>→</mo><mrow><mi>p</mi></mrow></mover><mn mathvariant="normal">0</mn><mtext> </mtext></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20930994&type= 61.12933731 7.61999989 引理2 设 <math id="M113"><mi mathvariant="bold-italic">ξ</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931010&type= 27.34733391 4.65666676 是包含 <math id="M114"><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931025&type= 1.86266661 2.79399991 个1和 <math id="M115"><mi>n</mi><mo>-</mo><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931037&type= 8.55133343 2.79399991 个0的随机变量，并且每一个置换都是等可能的。设 <math id="M116"><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931051&type= 32.85066605 4.48733330 是 <math id="M117"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931068&type= 8.97466660 3.55599999 上独立同分布的实值随机过程，并且 <math id="M118"><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931083&type= 29.04066467 4.65666676 ， <math id="M119"><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">{</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931096&type= 27.85533524 4.48733330 ， <math id="M120"><mi mathvariant="normal">V</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mo stretchy="false">{</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931107&type= 27.77066803 4.48733330 。设 <math id="M121"><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931120&type= 36.40666580 4.48733330 和 <math id="M122"><mi mathvariant="bold-italic">ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931134&type= 1.86266661 3.64066648 相互独立， <math id="M123"><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931146&type= 8.29733276 4.48733330 几乎所有轨道都有有限变差，则 <math id="M124"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931158&type= 40.63999939 8.38199997 在 <math id="M125"><msup><mrow><mi mathvariant="normal">𝓁</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msup><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931171&type= 13.20800018 3.80999994 上弱收敛到一个零均值的高斯过程。因此， <math id="M126"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931181&type= 38.69266510 8.38199997 依概率收敛到0，对 t 一致成立。 在下面定理的证明过程中， <math id="M127"><msub><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931188&type= 11.26066685 4.65666676 和 <math id="M128"><msub><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931200&type= 11.17599964 4.91066647 的渐近性质起着非常重要的作用。 <math display="block" id="M129"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo></mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931209&type= 134.78933716 8.63599968 <math display="block" id="M130"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msubsup><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><msub><mrow/><mrow><mi>k</mi></mrow></msub></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo></mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931223&type= 143.67933655 8.63599968 （3）和（4）式可由引理2、Glivenko-Cantelli引理和泛函Delta方法得到。 定理1的证明。我们首先证明 <math id="M131"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931232&type= 1.94733346 3.64066648 的相合性。根据 Inverse Function Theorem ［ 15 15 ］ ，要证明估计量的相合性，需要验证以下几点成立： （i） <math id="M132"><mo>∂</mo><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931241&type= 18.11866570 3.89466691 存在，并且在 <math id="M133"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931256&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931252&type= 3.04800010 3.72533321 的一个开邻域 <math id="M134"><mi mathvariant="normal">ℬ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931295&type= 2.45533323 2.87866688 上连续。 （ii） 当 <math id="M135"><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931270&type= 10.24466610 2.79399991 时，矩阵 <math id="M136"><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>∂</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><msup><mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931278&type= 29.29466629 4.82600021 正定的概率趋于1。 （iii） 对 <math id="M137"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931287&type= 3.04800010 3.72533321 的一个开邻域 <math id="M138"><mi mathvariant="normal">ℬ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931295&type= 2.45533323 2.87866688 内， <math id="M139"><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>∂</mo><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>/</mo><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931305&type= 27.17799950 3.89466691 依概率收敛到 <math id="M140"><mi mathvariant="bold-italic">A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931312&type= 2.79399991 2.79399991 一致成立。 （iv） 估计函数是渐近无偏的，即 <math id="M141"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>∂</mo><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><msup><mrow><mo>→</mo></mrow><mrow><mi>p</mi></mrow></msup><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931320&type= 23.87599945 3.80999994 。 我们可以计算 <math display="block" id="M142"><mfrac><mrow><mo>∂</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfrac><mo>=</mo><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mfrac><mrow><mo>∂</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931331&type= 168.14799500 8.72066593 <math display="block" id="M143"><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931341&type= 88.30733490 9.14400005 <math display="block" id="M144"><mo>-</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi>W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931350&type= 162.05200195 8.38199997 <math display="block" id="M145"><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931360&type= 82.55000305 9.14400005 利用大数定律，容易证明上式右边第二项收敛到0。上式右边第一项可以分解为 <math display="block" id="M146"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931370&type= 155.19400024 8.38199997 <math display="block" id="M147"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931380&type= 96.68933105 8.46666718 <math display="block" id="M148"><mo stretchy="false">(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931392&type= 85.42866516 4.91066647 <math display="block" id="M149"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931402&type= 101.34600067 8.46666718 <math display="block" id="M150"><mo stretchy="false">(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931413&type= 85.42866516 4.91066647 <math display="block" id="M151"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931421&type= 108.79666138 8.38199997 <math display="block" id="M152"><mo stretchy="false">(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931435&type= 85.42866516 4.91066647 <math display="block" id="M153"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931444&type= 108.62733459 8.38199997 <math display="block" id="M154"><mi>n</mi><mo stretchy="false">(</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931457&type= 82.88867188 4.91066647 应用引理2和文献［ 14 14 ］中的定理3.2，可以证明： <math id="M155"><mfrac><mrow><mo>∂</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></mrow></mfrac><mo>→</mo><mi mathvariant="bold-italic">A</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931471&type= 26.41600037 8.04333401 当 <math id="M156"><mi>n</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931486&type= 10.24466610 2.79399991 。因此，（ii）和（iii）是满足的。 对于（iv），通过简单的计算，可以把 <math id="M157"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931496&type= 16.42533302 4.65666676 分解为下面4个部分 <math display="block" id="M158"><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931507&type= 100.83799744 39.11600113 对固定 t ， <math id="M159"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931538&type= 64.51599884 8.38199997 是 n 个零均值独立随机变量的和， <math id="M160"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931530&type= 59.68999863 8.38199997 是有界变差函数，可以用两个单调函数的差表示。根据文献［ 16 16 ］中例2.11.16，可得到： <math id="M161"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931538&type= 64.51599884 8.38199997 弱收敛到一个零均值的高斯过程。 利用正则条件（C3）， <math id="M162"><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931548&type= 14.39333248 4.48733330 是一个有界变差函数，从而能表示成两个单调函数的差。根据引理2， （5）式右边第二项和第四项都收敛到0。 下面对（5）式右边的第三项进行如下分解 <math display="block" id="M163"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931556&type= 93.64133453 8.38199997 <math display="block" id="M164"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931563&type= 102.95466614 8.46666718 <math display="block" id="M165"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msubsup><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msup><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931568&type= 114.89267731 8.38199997 <math display="block" id="M166"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931575&type= 108.28866577 8.46666718 <math display="block" id="M167"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msup><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo></mrow><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msubsup><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931584&type= 115.40067291 8.38199997 借助（3）式、引理2和简单代数计算，等 式（6） 式（6） 右边第二项渐近等于 <math display="block" id="M168"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931591&type= 121.24267578 8.72066593 同理，利用（4）和引理2，等 式（6） 式（6） 右边的第四项渐近等于 <math display="block" id="M169"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>×</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931599&type= 159.93533325 8.72066593 因此，有 <math display="block" id="M170"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931608&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931607&type= 96.09667206 8.38199997 <math display="block" id="M171"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931614&type= 125.64533997 8.72066593 <math display="block" id="M172"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931624&type= 114.13066101 8.46666718 <math display="block" id="M173"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931632&type= 113.36865997 8.72066593 在正则条件下，（7）式右边的第一项渐近均值为零，协方差矩阵为 <math id="M174"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">{</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mo>⊗</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931639&type= 81.11067200 8.38199997 的正态分布 ［ 17 17 ］ 。 利用引理2，容易证明（7）式右边第二项和第三项分别都渐近于零均值的正态分布，其协方差矩阵分别为： <math id="M175"><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow><mrow><mi>α</mi></mrow></mfrac><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931644&type= 20.06599998 7.53533268 和 <math id="M176"><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow><mrow><mi>α</mi></mrow></mfrac><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mi>P</mi></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931651&type= 85.76733398 8.46666718 。 因此， <math id="M177"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931673&type= 16.42533302 4.65666676 依分布收敛到零均值的正态分布，其协方差矩阵为 <math display="block" id="M178"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi></mrow><mrow><mi>α</mi></mrow></mfrac><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">)</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mi>P</mi></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931667&type= 99.73733521 8.46666718 由上面结论知 <math id="M179"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931673&type= 16.42533302 4.65666676 依概率收敛到0，从而证明了定理1。 定理2的证明。我们首先进行如下分解 <math display="block" id="M180"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">{</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">{</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931680&type= 125.13733673 5.50333309 利用 <math display="block" id="M181"><mfrac><mrow><mo>∂</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">θ</mi></mrow></mfrac><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">β</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi mathvariant="bold-italic">γ</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo>-</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931686&type= 124.45999908 38.77733231 和泰勒展开式，可以得到（8）式的第一项为 <math display="block" id="M182"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931692&type= 90.00066376 5.41866684 对 <math id="M183"><mi>t</mi><mo>∈</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931698&type= 13.97000027 3.55599999 一致成立，其中 <math display="block" id="M184"><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>u</mi></mtd></mtr><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931701&type= 87.20666504 20.82799911 对于（8）式的第二项，我们有 <math display="block" id="M185"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931706&type= 95.84266663 17.94933319 <math display="block" id="M186"><mo>+</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931712&type= 57.82733154 17.94933319 应用文献［ 5 5 ］的方法，可以证明 <math display="block" id="M187"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mo stretchy="false">(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931720&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931717&type= 136.56732178 17.94933319 利用定理1的证明过程，类似地可以推导出 <math display="block" id="M188"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mrow><mrow><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931723&type= 117.09400177 17.94933319 <math display="block" id="M189"><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>-</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931728&type= 131.48733521 8.97466660 <math display="block" id="M190"><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></mrow><mo>×</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931734&type= 93.64133453 8.72066593 <math display="block" id="M191"><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931740&type= 88.98467255 8.80533314 结合（9），（11）和（12）式，可得 <math display="block" id="M192"><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>Λ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931746&type= 36.15266418 5.41866684 <math display="block" id="M193"><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mover><mrow><mi>α</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfrac><mo stretchy="false">)</mo></mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>o</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931752&type= 116.41667175 8.46666718 其中， <math display="block" id="M194"><msub><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo></mrow></mrow><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931758&type= 93.04866791 8.89000034 <math display="block" id="M195"><mo>+</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msubsup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931763&type= 63.83866882 8.63599968 <math display="block" id="M196"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></mrow><msub><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">R</mi></mrow></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">R</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>u</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931769&type= 136.90600586 9.05933285 <math display="block" id="M197"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>-</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo><mtext> </mtext><mo>×</mo><mfrac><mrow><mi mathvariant="normal">d</mi><mi>u</mi></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931773&type= 159.34268188 8.97466660 <math display="block" id="M198"><msubsup><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mover><mrow><mi>q</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931780&type= 136.31333923 8.89000034 <math display="block" id="M199"><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow></mfrac></mrow><mo stretchy="false">(</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931786&type= 150.70666504 8.80533314 <math display="block" id="M200"><msubsup><mrow><mi>ζ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo></mrow><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow><mrow><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931792&type= 106.00266266 17.44133186 令 <math id="M201"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931797&type= 66.37866211 4.74133301 ，其中 <math id="M202"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931803&type= 41.82533264 8.38199997 ； <math id="M203"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931808&type= 45.46599960 4.74133301 <math id="M204"><mo>…</mo><mo>,</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931817&type= 21.25133324 4.74133301 ,其中 <math id="M205"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>/</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><msub><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931823&type= 58.41999817 8.38199997 ； <math id="M206"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931829&type= 66.29399872 4.74133301 ,其中 <math id="M207"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></mrow></msup><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>ν</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup></mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931835&type= 42.07933044 8.38199997 , <math id="M208"><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931840&type= 22.35199928 3.21733332 。 根据定理2 ［ 17 17 ］ ， <math id="M209"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931932&type= 11.68400002 3.80999994 和 <math id="M210"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931936&type= 11.76866722 3.80999994 分别弱收敛到 <math id="M211"><mi>D</mi><msup><mrow><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></mrow><mrow><mi>K</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931857&type= 14.47799969 3.80999994 上零均值的高斯过程 <math id="M212"><msup><mrow><mi mathvariant="bold-italic">𝒲</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931862&type= 66.80200195 4.65666676 和 <math id="M213"><msup><mrow><mi mathvariant="bold-italic">𝒲</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931867&type= 67.22533417 4.65666676 ，并且 <math id="M214"><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931872&type= 12.36133385 4.65666676 和 <math id="M215"><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931877&type= 12.36133385 4.65666676 的协方差函数为 <math id="M216"><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>ν</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>ν</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931883&type= 35.72933197 4.48733330 ， <math id="M217"><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931888&type= 12.44600010 4.65666676 和 <math id="M218"><msubsup><mrow><mi mathvariant="italic">𝒲</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931894&type= 12.44600010 4.65666676 的协方差函数为 <math id="M219"><mi>E</mi><mo stretchy="false">{</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931898&type= 36.74533463 4.65666676 。 类似地， <math id="M220"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931941&type= 11.68400002 3.80999994 弱收敛到一个零均值的高斯过程， <math id="M221"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931905&type= 12.61533356 4.65666676 和 <math id="M222"><msubsup><mrow><mi>W</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931911&type= 12.61533356 4.65666676 的协方差函数为 <math display="block" id="M223"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><mtext> </mtext><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">v</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931915&type= 129.37066650 8.46666718 <math display="block" id="M224"><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931920&type= 22.18266678 4.48733330 <math display="block" id="M225"><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">v</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931924&type= 119.97266388 8.46666718 <math display="block" id="M226"><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfrac><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">v</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>m</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>K</mi></mrow></munderover></mstyle><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>k</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfrac><mo>×</mo><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">C</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">v</mi><mo stretchy="false">{</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>m</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msubsup><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>m</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>m</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">Γ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931928&type= 252.98399353 8.46666718 根据条件期望的性质， <math id="M227"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931932&type= 11.68400002 3.80999994 ， <math id="M228"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931936&type= 11.76866722 3.80999994 和 <math id="M229"><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931941&type= 11.68400002 3.80999994 相互独立。因此， <math id="M230"><mi mathvariant="bold-italic">W</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931947&type= 57.48867035 3.80999994 弱收敛到一个零均值的高斯过程： <math id="M231"><mi mathvariant="bold-italic">𝒲</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">𝒲</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">𝒲</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">𝒲</mi></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931952&type= 59.68999863 3.80999994 。 3　 数值模拟 1 本节将通过数值模拟来检验所提出估计的性质。为了便于比较，把与时间无关的权重计算的参数用 <math id="M232"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931956&type= 2.79399991 4.57200003 表示，把与时间相关的权重计算的参数用 <math id="M233"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931961&type= 3.47133350 4.57200003 表示。在数值模拟中，本文考虑两种类型的疾病，即 K =2。协变量 <math id="M234"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931965&type= 15.23999977 4.14866638 和 <math id="M235"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931977&type= 4.91066647 3.80999994 是来自成功概率为0.5的Bernoulli分布。在给定 <math id="M236"><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mtext> </mtext><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931973&type= 15.23999977 4.14866638 和 <math id="M237"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931979&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931977&type= 4.91066647 3.80999994 的条件下，我们考虑失效时间（ <math id="M238"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931981&type= 9.39799976 4.14866638 ）由下面的联合生存函数 ［ 18 18 ］ 产生 <math display="block" id="M239"><mi>S</mi><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></munderover></mstyle><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi></mrow><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">0</mn><mi>k</mi></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><msub><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">}</mo><mo stretchy="false">)</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mrow><mi>κ</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mi>κ</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931987&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931985&type= 103.03933716 8.55133343 其中 <math id="M240"><mi>κ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20931989&type= 8.63599968 2.79399991 表示 <math id="M241"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932045&type= 3.72533321 3.72533321 和 <math id="M242"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932051&type= 3.72533321 3.72533321 的相关程度。 <math id="M243"><mi>κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932041&type= 1.86266661 2.79399991 的值越小，则表明 <math id="M244"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932045&type= 3.72533321 3.72533321 和 <math id="M245"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932051&type= 3.72533321 3.72533321 之间的相关性越强。 受Lin等 ［ 14 14 ］ 的启发，本文选取的 <math id="M246"><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932019&type= 13.03866673 4.48733330 的形式如下 <math display="block" id="M247"><mfenced separators="|"><mrow><mtable><mtr><mtd><msup><mrow><mi>g</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">β</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msup><mrow><mi>h</mi></mrow><mrow><mi>'</mi></mrow></msup><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mi>h</mi><mo stretchy="false">{</mo><msup><mrow><mi mathvariant="bold-italic">γ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>/</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mi mathvariant="bold-italic">γ</mi><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy="false">}</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">Z</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932025&type= 89.23867035 10.75266743 在数值模拟中，令 <math id="M248"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932030&type= 10.75266743 3.72533321 和 <math id="M249"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932034&type= 10.75266743 3.72533321 。 <math id="M250"><mi>κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932041&type= 1.86266661 2.79399991 的值取为0.10，0.80和1.25，相应的Kendall’s tau是0.83，0.38和0.29，这表明 <math id="M251"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932048&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932045&type= 3.72533321 3.72533321 和 <math id="M252"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932051&type= 3.72533321 3.72533321 的相关程度越来越弱。删失时间由均匀分布 <math id="M253"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>u</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932055&type= 9.14400005 3.55599999 产生，而 u 的值是根据数据删失的比例来确定的。另外，考虑疾病的发生率分别为 <math id="M254"><msub><mrow><mi>P</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">18</mn><mi mathvariant="normal">%</mi><mo>,</mo><mn mathvariant="normal">32</mn><mi mathvariant="normal">%</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932062&type= 28.78666496 4.65666676 和 <math id="M255"><msub><mrow><mi>P</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">26</mn><mi mathvariant="normal">%</mi><mo>,</mo><mn mathvariant="normal">41</mn><mi mathvariant="normal">%</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932068&type= 28.87133408 4.65666676 ，在广义case-cohort设计下，抽取的子列大小 <math id="M256"><mover><mrow><mi>n</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932073&type= 1.86266661 2.79399991 为300， <math id="M257"><mi>q</mi><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">0.5,0.5</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932079&type= 21.33600044 3.72533321 。全队列的 n =1 000，对每一种情形的参数估计，都重复进行1 000次模拟，数值模拟结果见 表1 表1 。 注： Bias、SE、SD和CR分别表示估计值与真值间的偏差、标准差的平均估计量、经验的标准误差和风险率的95%的经验覆盖概率 2 q 2 <math id="M260"><mi>κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932091&type= 1.60866666 2.37066650 2 Para. 7 <math id="M261"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932399&type= 2.45533323 3.89466691 7 <math id="M262"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932423&type= 3.13266683 3.89466691 Bias 2 SE 2 SD 2 CR Bias 2 SE 2 SD 2 CR 14 P D =[18%,32%] 0.10 β =0 2 <math id="M263"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.009 0 2 0.172 7 2 0.174 7 0.952 2 0.009 3 2 0.173 4 2 0.172 1 0.955 γ =0 2 0.005 5 2 0.117 7 2 0.117 0 0.950 2 <math id="M264"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.002 1 2 0.118 8 2 0.119 6 0.935 0.80 β =0 2 0.006 8 2 0.173 5 2 0.171 9 0.947 2 0.004 6 2 0.173 0 2 0.180 1 0.941 γ =0 2 <math id="M265"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.000 1 2 0.118 5 2 0.121 0 0.946 2 0.003 7 2 0.118 4 2 0.120 5 0.952 1.25 β =0 2 <math id="M266"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.002 4 2 0.173 5 2 0.172 3 0.949 2 0.000 7 2 0.173 0 2 0.175 9 0.955 γ =0 2 <math id="M267"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.001 6 2 0.117 8 2 0.115 6 0.953 2 <math id="M268"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.008 1 2 0.118 2 2 0.117 2 0.951 [0.5,0.5] 0.10 β =0 2 0.003 2 2 0.196 5 2 0.201 2 0.948 2 0.001 7 2 0.195 8 2 0.200 7 0.947 γ =0 2 <math id="M269"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.005 4 2 0.134 2 2 0.132 7 0.954 2 <math id="M270"><mo>-</mo></math> 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2 0.245 6 0.950 γ =0.5 2 <math id="M296"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.025 0 2 0.135 5 2 0.138 3 0.937 2 <math id="M297"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.023 7 2 0.136 0 2 0.141 1 0.940 1.25 β =0.5 2 <math id="M298"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.032 7 2 0.247 2 2 0.245 8 0.950 2 <math id="M299"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.004 7 2 0.247 8 2 0.251 7 0.934 γ =0.5 2 <math id="M300"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.024 8 2 0.135 1 2 0.134 7 0.947 2 <math id="M301"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932354&type= 3.72533321 2.79399991 0.017 5 2 0.136 4 2 0.136 9 0.950 从 表1 表1 可以看出，两种估计 <math id="M302"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932371&type= 2.79399991 3.72533321 和 <math id="M303"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932380&type= 3.47133350 3.72533321 的模拟结果是相似的。具体来说，在不同的疾病发生率下， <math id="M304"><mi>β</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>γ</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932386&type= 19.13466644 3.80999994 时的估计都是无偏的，并且它们的标准差的估计效果都很好，95%的经验覆盖概率CR值介于93%~96%之间，这是合理的。 另外，当 <math id="M305"><mi>q</mi><mo>=</mo><mo stretchy="false">[</mo><mn mathvariant="normal">1,1</mn><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932392&type= 15.23999977 3.72533321 时，广义case-cohort设计下多类型事件数据的加乘风险回归模型就退化为一般case-cohort设计下的情形。因此， 表1 表1 实际上给出了广义case-cohort设计和一般case-cohort设计下的参数估计值结果比较。显然，从 表1 表1 可以看出，广义case-cohort设计下的结果与一般case-cohort设计下的结果类似。但当疾病事件发生率较高时，广义case-cohort设计下得到的结果是非常好的。因此，广义case-cohort设计在疾病事件发生率较高时更加具有应用性。 4　 实例分析 1 为了对比分析广义case-cohort设计的实用性，本节将所提方法应用到澳大利亚Busselton的健康调查数据 ［ 19 19 ］ 上，比较采用该设计方法与一般case-cohort设计的应用效果。 由于血液样本的收集与保存费用昂贵，为了降低成本，在调查中实施了case-cohort抽样设计，化验的活性血清样本总共有626份，包含子列中的450人以及子列外发病者176人（冠心病113人，中风39人，两种疾病都发生的24人）。对子列外的176名发病个体按比例 q =［0.5，0.5］抽取部分个体收集协变量。这样，子列外共收集88名发病个体的协变量信息。在广义case-cohort设计下，收集到的样本总共有538份。 表2 表2 和 表3 表3 分别给出了 q =［1，1］和 q =［0.5，0.5］时，广义case-cohort设计下的模型和一般case-cohort设计下的模型的应用结果。结果表明，两种模型结果类似，但在广义case-cohort设计下风险率的95%的经验覆盖概率区间的精度有所下降。 注： Est、Std、HR和CR分别表示估计值、标准差的估计值、风险率估计值和风险率的95%的经验覆盖概率 2 参数 4 <math id="M306"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932399&type= 2.45533323 3.89466691 4 <math id="M307"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932423&type= 3.13266683 3.89466691 Est Std HR CR Est Std HR CR log(ferritin) on CHD 0.095 1 0.098 9 1.099 8 (0.906 0, 1.335 0) 0.095 0 0.098 9 1.099 6 (0.905 8, 1.335 0) log(ferritin) on stroke 0.197 9 0.142 1 1.218 8 (0.922 4, 1.610 4) 0.198 5 0.141 8 1.219 5 (0.923 7, 1.610 2) Age 0.072 9 0.007 1 1.075 6 (1.060 8, 1.090 7) 0.072 9 0.007 1 1.075 7 (1.060 8, 1.090 7) Triglyceides 0.265 4 0.049 2 1.303 9 (1.184 2, 1.435 8) 0.265 5 0.049 0 1.304 1 (1.184 6, 1.435 7) Diabetes treatment 0.002 0 0.008 7 1.002 0 (0.985 1, 1.019 1) 0.002 0 0.008 6 1.002 0 (0.985 3, 1.018 9) Blood pressure treatment 0.003 0 0.002 1 1.003 0 (0.998 8, 1.007 2) 0.003 0 0.002 1 1.003 0 (0.998 8, 1.007 2) 注： Est、Std、HR和CR分别表示估计值、标准差的估计值、风险率估计值和风险率的95%的经验覆盖概率 2 参数 4 <math id="M308"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932419&type= 2.45533323 3.89466691 4 <math id="M309"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">I</mi><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=20932423&type= 3.13266683 3.89466691 Est Std HR CR Est Std HR CR log(ferritin) on CHD 0.094 0 0.124 9 1.098 5 (0.859 9, 1.403 4) 0.093 8 0.123 4 1.098 4 (0.862 5,1.398 8) log(ferritin) on stroke 0.205 6 0.181 6 1.228 2 (0.860 3, 1.753 5 ) 0.206 0 0.181 1 1.228 7 (0.861 6,1.752 2) Age 0.067 9 0.008 7 1.070 2 (1.052 2 , 1.088 5) 0.068 0 0.008 6 1.070 4 (1.052 5,1.088 5) Triglyceides 0.233 9 0.085 5 1.263 6 (1.068 5 , 1.494 1) 0.234 9 0.084 5 1.264 8 (1.071 7,1.492 7) Diabetes treatment 0.000 9 0.008 6 1.000 9 (0.984 1, 1.018 0 ) 0.000 8 0.008 2 1.000 8 (0.984 9,1.017 0) Blood pressure treatment 0.003 0 0.002 1 1.003 0 (0.998 8, 1.007 2) 0.002 9 0.002 0 1.002 9 (0.998 9,1.006 9) 5　 结 语 1 本文研究广义case-cohort设计下多类型事件数据的加乘风险回归模型。利用逆概率加权的思想，给出加权估计方程，并证明参数估计的相合性和渐近正态性。数值模拟结果说明所提出方法的可行性与有效性。在数值模拟中，本文还使用了两类权重函数：与时间相关的权重和与时间无关的权重，并与一般case-cohort设计下的参数估计值结果进行比较。数值结果表明：1） 与时间相关权重的估计效果略微高于与时间无关权重的估计效果，但效果并不明显；2） 广义case-cohort设计下的结果与一般case-cohort设计下的结果类似，但当疾病事件发生率较高时，广义case-cohort设计更加具有应用性。最后，将本文所提模型应用到澳大利亚Busselton的健康调查数据中上，实例分析表明估计结果良好，但在广义case-cohort设计下风险率的95%的经验覆盖概率区间的精度有所下降。 参考文献 PRENTICE R L . 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DOI:10.14188/j.1671-8836.2020.0182 安徽省高等学校自然科学研究重点项目（KJ2018A0390） 2020-07-18 2021-10-24 2021-10-12 武汉大学学报（理学版） 5 67