基于混合Gamma分布单个观测的可靠性函数无偏估计

基于混合Gamma分布单个观测的可靠性函数无偏估计 Unbiased Estimation of Reliability Function Based on One Single Observation from a Mixture of Gamma Distributions 1 first-author 王博锐，男，硕士生，从事混合分布理论研究。E-mail：1693864298@qq.com 王博锐，男，硕士生，从事混合分布理论研究。E-mail：1693864298@qq.com 1693864298@qq.com 1 corresp E-mail：lqliu@whu.edu.cn E-mail：lqliu@whu.edu.cn lqliu@whu.edu.cn 武汉大学 数学与统计学院，湖北 武汉430072 武汉大学 数学与统计学院，湖北 武汉430072 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China 针对小样本甚至单样本情况下的疲劳寿命分布特征研究，给出了混合Gamma分布的概率密度函数可定义的充要条件，基于混合Gamma分布的单个观测值，得到了Gamma分布的可靠性函数的两个无偏估计，并讨论了所得无偏估计的适当性， 最后进行了数值模拟验证。 In order to study the characteristics of fatigue life distributions in small sample or one sample，we gave the sufficient and necessary condition for the definition of probability density function of mixed Gamma distribution. Then we obtained two unbiased estimations of the reliability function based on one single observation from a mixture of two Gamma distributions. Next we discussed the appropriateness of the obtained unbiased estimations. Finally, we did some simulations. 混合分布 污染数据 可靠性函数 Gamma分布 无偏估计 mixture distribution contaminated data reliability function Gamma distribution unbiased estimation 0　 引 言 1 在传统的研究中，人们一般首先假设样本数据来自某个特定的分布总体，然后基于这个总体进行后续的统计推断。但随着科技发展，我们所面临的数据越来越复杂。在很多情形下，样本数据在不同局部可能有不同的特性，单一的参数分布族无法确切地描述观测数据，人们想到用混合分布模型对复杂数据进行统计分析。混合分布模型最早可追溯到Newcomb [ 1 1 ] 和Pearson [ 2 2 ] 的相关工作。近三十几年来，混合分布模型发展迅速，应用广泛，尤其是在生物分析、资源评价、金融股票等领域 [ 3 3 ] 。比较典型的例子有 [ 4 4 ] :l) 渔业研究中，鱼的体长可以度量，但其性别和年龄不易确定，需要研究如何由体长数据确定不同年龄组和不同性别鱼群的比例；2) 沉积岩学中，一个沙样中颗粒大小的分布已知，但不知道其中包含有哪些不同种类的矿石；3) 医学诊断中可对一组病人做临床调查，但他们所患疾病属于哪一种类型是未知的；4) 在RFLP-QTL连锁分析中，可以知道每个个体的标记基因型及性状值，这个性状值是由三种类型的QTL基因型混合而成，并且个体的QTL基因型未知，如何由标记基因型的表型去推断QTL与标记之间的连锁以及QTL效应值。此外，混合分布模型在寿命数据分析 [ 5 5 ] 和污染数据分析中也扮演着重要角色。早在1952年，Davis [ 6 6 ] 就注意到在寿命实验中，元件寿命分布函数可能为两个分布函数的混合。 考虑独立同分布的随机变量列 <math id="M1"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420596&type= 15.24000072 3.97933340 ，具有分布函数 <math display="block" id="M2"><msub><mrow><mi>F</mi></mrow><mrow><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>α</mi><mo stretchy="false">)</mo><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>α</mi><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420598&type= 52.32399750 4.48733330 其中 <math id="M3"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420600&type= 21.75933456 4.48733330 都是分布函数。通常我们更关心 <math id="M4"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420602&type= 9.82133293 4.48733330 ，认为数据样本应该服从 <math id="M5"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420604&type= 9.82133293 4.48733330 ，但却受到了来自分布 <math id="M6"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420606&type= 9.82133293 4.48733330 的数据的污染，称 <math id="M7"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420608&type= 9.82133293 4.48733330 为主分布， <math id="M8"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420610&type= 1.94733334 2.62466669 为污染系数，它衡量数据受污染的程度。1964年Huber [ 7 7 ] 考虑了 <math id="M9"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420612&type= 9.82133293 4.48733330 为标准正态分布而 <math id="M10"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420614&type= 9.82133293 4.48733330 是一族关于原点对称的分布的情形。一个特殊的情况就是: <math id="M11"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420616&type= 9.82133293 4.48733330 和 <math id="M12"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420618&type= 9.82133293 4.48733330 是具有相同形式但含有不同参数的分布，即 <math id="M13"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><msub><mrow><mi>θ</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=3420621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420620&type= 57.99666595 4.48733330 [ 8 8 ] 。近十年来，删失数据和截尾数据 [ 9 9 ] 的发展已经取得了很大的成果，但是关于污染数据 [ 10 10 ] 的研究却很少。污染数据的处理越来越受到重视，有广泛应用背景。 污染数据服从混合分布，对于混合分布常用的参数估计方法有矩估计、极大似然估计等方法。但遗憾的是，在小样本场合，这些估计将会有较大的偏差和方差。小样本估计问题在社会经济系统、生物医学系统和航空航海系统等领域中是普遍存在的。在疲劳寿命实验中，随着科学技术的迅速发展，产品的可靠性越来越高。很多时候由于人力、财力等的限制，重复实验的代价昂贵，这时可以获取的实验数据是非常少的。如果依然采用极大似然估计等方法进行统计推断，一般无法达到期望的效果。因此，适用于小样本情况下的疲劳寿命分布特性研究是很有必要的。早在20世纪60年代，国外学者已经开始研究小样本问题，80至90年代国内外普遍开始重视小样本问题的研究。比小样本情况更极端的一种情形为单样本情况，即只根据一个观测样本进行相关的统计推断。单样本情形下的统计推断研究非常少。(1)式中，在 <math id="M14"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420622&type= 9.82133293 4.48733330 和 <math id="M15"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420624&type= 9.82133293 4.48733330 是具有相同形式但含有不同参数的指数分布的特殊情形下，2017年Nie等 [ 11 11 ] 考虑了基于混合指数分布的单个观测值对 <math id="M16"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420626&type= 9.82133293 4.48733330 的可靠性函数的无偏估计。他们假设随机变量 <math id="M17"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420628&type= 2.53999996 2.62466669 服从参数为 <math id="M18"><mi>λ</mi><mo stretchy="false">（</mo><mi>λ</mi><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">）</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420630&type= 12.86933327 3.47133350 的指数分布，记为 <math id="M19"><mi>T</mi><mo>∼</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420632&type= 14.30866623 3.55599999 ，则 <math id="M20"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420634&type= 2.53999996 2.62466669 的可靠性函数为 <math id="M21"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mfrac><mrow><mi>t</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mo stretchy="false">)</mo><mo>,</mo><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420636&type= 38.26933289 7.19666672 。两个指数分布的混合，记为 <math id="M22"><mi>p</mi><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>λ</mi><mo stretchy="false">)</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420638&type= 35.98333359 3.72533321 <math id="M23"><mi>p</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>λ</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420640&type= 28.44799995 3.64066648 。值得注意的是，有时 <math id="M24"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420642&type= 1.86266661 3.47133350 在[0,1]之间时无法满足实际情况的需要，因此这里的混合分布可以理解为广义的混合分布。在 <math id="M25"><mi>p</mi><mo>,</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420644&type= 5.41866684 3.55599999 已知， <math id="M26"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420646&type= 1.77800000 2.70933342 未知的情况下，Nie等基于混合指数分布 <math id="M27"><mi>p</mi><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>λ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420648&type= 33.78200150 3.72533321 的单个观测值首先给出了 <math id="M28"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420650&type= 7.45066643 3.55599999 的两个无偏估计，然后讨论了这两个无偏估计的适当性，最后给出了这两个无偏估计的方差的无偏估计。 寿命数据分析方法是进行可靠性工程研究的基础。常见系统中元件可靠性参数以及工程材料的疲劳寿命和强度分布都可以用Gamma分布很好地描述。指数分布、卡方分布、Erlang分布是Gamma分布的特例。自从Gamma分布被提出，各国研究者就对Gamma分布的参数估计问题产生了极大兴趣并为之付出了不懈努力。极大似然估计、矩估计、贝叶斯估计等为最常用的方法。由于Gamma分布在拟合数据方面比较灵活，适应性较强，因此Gamma分布被广泛地应用于众多统计应用中。常用的寿命概率分布描述寿命数据规律的能力都有限，而一个拟合能力更好的寿命概率分布模型可以让我们更精确的进行统计推断。混合Gamma分布的稠密性是指对于任何非负随机变量，我们总可以选择一个适当的混合Gamma分布把它拟合到任意需要的精度。因此，从理论上讲，选择混合Gamma分布拟合寿命数据总是合理的 [ 12 12 ] 。 本文假设随机变量 <math id="M29"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420652&type= 2.53999996 2.62466669 服从参数为 <math id="M30"><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420654&type= 8.88999939 3.89466667 的Gamma分布，记为 <math id="M31"><mi>T</mi><mo>∼</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420656&type= 20.91266632 3.89466667 ，其概率密度函数为 <math display="block" id="M32"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msup><mrow><mi>β</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>β</mi><mi>x</mi></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo><</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420658&type= 57.65800095 14.39333344 其中 <math id="M33"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420660&type= 8.55133343 2.79399991 是形状参数， <math id="M34"><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420662&type= 8.72066689 3.81000018 是尺度参数， <math id="M35"><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420664&type= 6.68866634 3.55599999 为Gamma函数。Gamma分布与很多分布都有关系:当 <math id="M36"><mi>α</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420666&type= 8.55133343 2.70933342 时，它是指数分布;当 <math id="M37"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420668&type= 1.94733334 2.62466669 为正整数时，它是Erlang分布;当 <math id="M38"><mi>β</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420670&type= 9.99066734 7.19666672 时，它是卡方分布。 <math id="M39"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420672&type= 2.53999996 2.62466669 的可靠性函数为 <math id="M40"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>,</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420674&type= 39.28533173 6.77333355 <math id="M41"><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420676&type= 7.95866680 3.30199981 。考虑两个Gamma分布的混合 <math id="M42"><mi>p</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420678&type= 50.79999924 4.14866638 ，其中 <math id="M43"><mi>p</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420680&type= 39.28533173 3.81000018 。 <math id="M44"><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420682&type= 20.06599998 3.72533321 或 <math id="M45"><mi>θ</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420684&type= 8.55133343 2.70933342 时混合Gamma分布会退化成单个Gamma分布，故本文假定 <math id="M46"><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420686&type= 20.06599998 3.72533321 且 <math id="M47"><mi>θ</mi><mo>≠</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420688&type= 8.55133343 3.04799986 。我们认为数据本身服从的分布为 <math id="M48"><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420690&type= 13.80066681 3.89466667 ，但受到了来自分布 <math id="M49"><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420692&type= 19.30399895 4.14866638 数据的影响，我们更想得到分布 <math id="M50"><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420694&type= 13.80066681 3.89466667 的可靠性函数 <math id="M51"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420696&type= 7.45066643 3.55599999 。在 <math id="M52"><mi>p</mi><mo>,</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420698&type= 5.41866684 3.55599999 已知， <math id="M53"><mi>α</mi><mo>,</mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420700&type= 5.67266655 3.72533321 未知的情况下，本文基于混合Gamma分布的单个观测值首先给出 <math id="M54"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420702&type= 7.45066643 3.55599999 的两个无偏估计，然后讨论了这两个无偏估计的适当性。假设 <math id="M55"><mi>p</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420704&type= 39.28533173 3.81000018 。 <math id="M56"><mi>p</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420706&type= 50.79999924 4.14866638 的“密度”记为 <math display="block" id="M57"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>p</mi><mfrac><mrow><msup><mrow><mi>β</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>β</mi><mi>x</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>β</mi><mo>⋅</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></mfrac><mi>x</mi></mrow></msup><mo>,</mo></mtd><mtd><mi>x</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>x</mi><mo><</mo><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420708&type= 113.62266541 14.39333344 概率密度函数应该具有非负性。由于此时 <math id="M58"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420710&type= 1.86266661 3.47133350 的范围并不仅限于 <math id="M59"><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=3420713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420712&type= 8.97466660 3.55599999 之间，所以此时 <math id="M60"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420714&type= 23.62199974 3.89466667 可能为负；只有当 <math id="M61"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420716&type= 23.62199974 3.89466667 非负时，它才成为概率密度函数。 1　 <math id="M62"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420718&type= 25.99266624 4.48733330 的非负性 1 下面根据 <math id="M63"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420720&type= 1.86266661 3.47133350 的取值范围分三种情况讨论 <math id="M64"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420722&type= 23.62199974 3.89466667 的非负性。 1)　 <math id="M65"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420724&type= 8.55133343 3.64066648 。 <math id="M66"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420726&type= 23.62199974 3.89466667 非负，即要求 <math display="block" id="M67"><mi>p</mi><mfrac><mrow><msup><mrow><mi>β</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>β</mi><mi>x</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>β</mi><mo>⋅</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></mfrac><mi>x</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420728&type= 77.38533020 8.29733276 上式等价于 <math display="block" id="M68"><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>β</mi><mi>x</mi><mo>≥</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi><msup><mrow><mi>θ</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420730&type= 36.91466904 8.29733276 解上述不等式得： <math id="M69"><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot><mo>≤</mo><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420732&type= 27.17800140 9.31333351 。因此，当 <math id="M70"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420734&type= 8.55133343 3.64066648 时， <math id="M71"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420736&type= 30.14133263 4.06400013 当且仅当 <math id="M72"><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot><mo>≤</mo><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420738&type= 27.17800140 9.31333351 。 2)　 <math id="M73"><mi>p</mi><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420740&type= 8.55133343 3.64066648 。 <math id="M74"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420742&type= 23.62199974 3.89466667 非负，即要求 <math display="block" id="M75"><mi>p</mi><mfrac><mrow><msup><mrow><mi>β</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>β</mi><mi>x</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>β</mi><mo>⋅</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>β</mi></mrow><mrow><mi>θ</mi></mrow></mfrac><mi>x</mi></mrow></msup><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420744&type= 77.38533020 8.29733276 上式等价于 <math display="block" id="M76"><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>β</mi><mi>x</mi><mo>≤</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi><msup><mrow><mi>θ</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420746&type= 36.91466904 8.29733276 解上述不等式得： <math id="M77"><mn mathvariant="normal">1</mn><mo><</mo><mi>θ</mi><mo>≤</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420748&type= 26.75466537 9.31333351 。因此，当 <math id="M78"><mi>p</mi><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420750&type= 8.55133343 3.64066648 时， <math id="M79"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420752&type= 30.14133263 4.06400013 当且仅当 <math id="M80"><mn mathvariant="normal">1</mn><mo><</mo><mi>θ</mi><mo>≤</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420754&type= 26.75466537 9.31333351 。 3)　 <math id="M81"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420756&type= 15.15533447 3.64066648 。显然 <math id="M82"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420758&type= 30.14133263 4.06400013 。 综上所述， <math id="M83"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420760&type= 23.62199974 3.89466667 成为概率密度函数的充要条件为下面三种情况之一成立: (d1)　 <math id="M84"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420762&type= 8.55133343 3.64066648 且 <math id="M85"><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot><mo>≤</mo><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420764&type= 27.17800140 9.31333351 ; (d2)　 <math id="M86"><mi>p</mi><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420766&type= 8.55133343 3.64066648 且 <math id="M87"><mn mathvariant="normal">1</mn><mo><</mo><mi>θ</mi><mo>≤</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420768&type= 26.75466537 9.31333351 ; (d3) <math id="M88"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420770&type= 15.15533447 3.64066648 。 2　 可靠性函数的两个无偏估计 1 假设 <math id="M89"><mi>p</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420772&type= 18.37266731 3.64066648 已知， <math id="M90"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420774&type= 19.13466644 3.81000018 未知。设 <math id="M91"><mi>T</mi><mo>∼</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420776&type= 20.91266632 3.89466667 ， <math id="M92"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420778&type= 2.53999996 2.62466669 的概率密度函数为 <math id="M93"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420780&type= 16.00200081 3.89466667 ，可靠性函数为 <math id="M94"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420782&type= 48.76799774 6.77333355 。易知 <math id="M95"><mi>R</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420784&type= 14.47799969 3.55599999 。若条件(d1)、(d2)、(d3)之一成立，则 <math id="M96"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420786&type= 23.62199974 3.89466667 是混合Gamma分布 <math id="M97"><mi>p</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420788&type= 50.79999924 4.14866638 的概率密度函数。假设 <math id="M98"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420790&type= 2.62466669 2.62466669 是来自 <math id="M99"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420792&type= 23.62199974 3.89466667 的单个样本，我们的目的是基于单个观测 <math id="M100"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420794&type= 2.62466669 2.62466669 找到 <math id="M101"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420796&type= 18.96533394 3.55599999 的无偏估计。 假设 <math id="M102"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420798&type= 11.68400002 3.72533321 是定义在 <math id="M103"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420800&type= 24.21466637 3.55599999 上的实值函数且关于 <math id="M104"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420802&type= 2.03200006 2.62466669 左连续，下面的定理给出 <math id="M105"><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420804&type= 12.53066635 3.72533321 是 <math id="M106"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420806&type= 7.45066643 3.55599999 的无偏估计的必要条件。 定理1 假设条件(d1)、(d2)、(d3)之一成立， <math id="M107"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420808&type= 2.62466669 2.62466669 是来自 <math id="M108"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420810&type= 23.62199974 3.89466667 的单个样本。若 <math id="M109"><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><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=3420813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420812&type= 33.35866928 3.72533321 <math id="M110"><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420814&type= 10.15999985 2.96333337 ，则 <math id="M111"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420816&type= 11.68400002 3.72533321 满足 <math display="block" id="M112"><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo> </mo><mo>∀</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420818&type= 78.57066345 3.72533321 证 若 <math id="M113"><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420820&type= 43.51866913 3.72533321 ，则 <math display="block" id="M114"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><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=3420823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420822&type= 80.09466553 6.77333355 <math display="block" id="M115"><mtable columnalign="left"><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mrow></mtd></mtr><mtr><mtd><mi>p</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mrow></mtd></mtr><mtr><mtd><mi>p</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mrow></mtd></mtr><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420824&type= 101.93866730 32.59666443 从而 <math id="M116"><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420826&type= 10.15999985 2.96333337 ，有 <math display="block" id="M117"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>β</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext>  </mtext><mo>∀</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420828&type= 117.94066620 8.12800026 由拉氏变换的唯一性， <math id="M118"><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420830&type= 10.15999985 2.96333337 ，有 <math display="block" id="M119"><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>,</mo><mtext> </mtext><mi mathvariant="normal">a</mi><mo>.</mo><mo> </mo><mo> </mo><mi mathvariant="normal">e</mi><mo>.</mo><mo> </mo><mtext> </mtext><mo> </mo><mo> </mo><mo> </mo><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420832&type= 87.20666504 3.72533321 又 <math id="M120"><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420834&type= 13.46199894 3.55599999 和 <math id="M121"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420836&type= 11.68400002 3.72533321 是关于 <math id="M122"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420838&type= 2.03200006 2.62466669 左连续的，所以 <math display="block" id="M123"><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mo stretchy="false">(</mo><mi>p</mi><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>，</mo><mo>∀</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420840&type= 86.36000061 3.72533321 证毕。 引理1 假设条件(d1)、(d2)、(d3)之一成立， <math id="M124"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420842&type= 2.62466669 2.62466669 是来自 <math id="M125"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420844&type= 23.62199974 3.89466667 的单个样本。若 <math id="M126"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420846&type= 11.68400002 3.72533321 满足(2)式，则有 <math display="block" id="M127"><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo> </mo><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420848&type= 52.57799911 3.97933340 证 若 <math id="M128"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420850&type= 11.68400002 3.72533321 满足(2)式，则有 <math display="block" id="M129"><mtable columnalign="left"><mtr><mtd><mi>E</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>X</mi><mo>;</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 mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi></mrow><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mtd></mtr><mtr><mtd><mi>p</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi></mrow><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>g</mi></mrow><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mtd></mtr><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo></mrow><mi>p</mi><mi>g</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo></mtd></mtr><mtr><mtd><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>I</mi></mrow><mo stretchy="false">(</mo><mi>θ</mi><mi>x</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>I</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420852&type= 115.40067291 32.59666443 证毕。 形式上定义 <math display="block" id="M130"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≜</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">（</mo><mi>x</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>k</mi></mrow></msup><mi>t</mi><mo stretchy="false">）</mo><mo>,</mo><mtext> </mtext><mi>x</mi><mo>,</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420855&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420854&type= 70.44266510 9.82133293 <math display="block" id="M131"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≜</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">（</mo><mi>x</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">）</mo><mo>,</mo><mtext> </mtext><mi>x</mi><mo>,</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420856&type= 84.83599854 9.99066734 下面的引理给出 <math id="M132"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420858&type= 12.95399952 4.65666676 和 <math id="M133"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420860&type= 12.95399952 4.65666676 收敛的充要条件。 引理2 <math id="M134"><mi>p</mi><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420862&type= 9.82133293 7.19666672 或 <math id="M135"><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420864&type= 8.55133343 2.79399991 是 <math id="M136"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420866&type= 12.95399952 4.65666676 收敛的充要条件; <math id="M137"><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420868&type= 9.82133293 7.19666672 或 <math id="M138"><mi>θ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420870&type= 8.55133343 2.79399991 是 <math id="M139"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420872&type= 12.95399952 4.65666676 收敛的充要条件。 证 首先分情况讨论 <math id="M140"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420874&type= 12.95399952 4.65666676 的收敛性。 1) <math id="M141"><mi>p</mi><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420876&type= 9.82133293 7.19666672 。由于 <math id="M142"><mfenced open="|" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420878&type= 18.96533394 9.05933285 ，因此 <math id="M143"><mo stretchy="false">|</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi><mo>-</mo><mo stretchy="false">|</mo><mi>p</mi><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=3420881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420880&type= 63.75400162 9.82133293 ，此时 <math id="M144"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420882&type= 12.95399952 4.65666676 收敛。 2) <math id="M145"><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420884&type= 8.55133343 2.79399991 。若 <math id="M146"><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420886&type= 14.56266594 3.30199981 ，则 <math id="M147"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420888&type= 19.47333336 4.65666676 ;若 <math id="M148"><mn mathvariant="normal">0</mn><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420890&type= 14.64733315 2.79399991 ，令 <math id="M149"><mi>N</mi><mo>=</mo><mfenced open="⌈" close="⌉" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>t</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420892&type= 27.09333420 8.29733276 ，则 <math id="M150"><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>N</mi></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420894&type= 30.64933395 3.55599999 ，故 <math id="M151"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420896&type= 39.62399673 9.82133293 收敛。 3) <math id="M152"><mi>p</mi><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420898&type= 9.82133293 7.19666672 且 <math id="M153"><mi>θ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420900&type= 8.55133343 2.79399991 。若 <math id="M154"><mn mathvariant="normal">0</mn><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420903&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420902&type= 14.64733315 2.79399991 ，则 <math id="M155"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420904&type= 39.62399673 9.82133293 发散;若 <math id="M156"><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420906&type= 14.56266594 3.30199981 ， <math id="M157"><mi>N</mi><mo>=</mo><mfenced open="⌊" close="⌋" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>t</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420908&type= 20.06599998 8.29733276 ，则 <math id="M158"><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>N</mi></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420910&type= 30.64933395 3.55599999 ，故 <math id="M159"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420913&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420912&type= 43.85733414 9.82133293 发散。 综上可知， <math id="M160"><mi>p</mi><mo>></mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420914&type= 9.82133293 7.19666672 或 <math id="M161"><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420916&type= 8.55133343 2.79399991 是 <math id="M162"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420918&type= 12.95399952 4.65666676 收敛的充要条件。接下来分情况讨论 <math id="M163"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420920&type= 12.95399952 4.65666676 的收敛性。 1) <math id="M164"><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420922&type= 9.82133293 7.19666672 。由于 <math id="M165"><mfenced open="|" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420924&type= 25.39999962 9.22866726 ，因此 <math id="M166"><mo stretchy="false">|</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo>-</mo><mo stretchy="false">|</mo><mi>p</mi><mo stretchy="false">|</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420926&type= 76.36933136 9.99066734 ，此时 <math id="M167"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420928&type= 12.95399952 4.65666676 收敛。 2) <math id="M168"><mi>θ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420930&type= 8.55133343 2.79399991 。若 <math id="M169"><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>θ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420932&type= 16.34066582 3.30199981 ，则 <math id="M170"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420935&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420934&type= 19.47333336 4.65666676 ;若 <math id="M171"><mn mathvariant="normal">0</mn><mo><</mo><mi>θ</mi><mi>t</mi><mo><</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420936&type= 16.42533302 2.79399991 ，令 <math id="M172"><mi>N</mi><mo>=</mo><mfenced open="⌈" close="⌉" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420938&type= 27.09333420 8.29733276 ，则 <math id="M173"><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420940&type= 25.23066711 3.55599999 ，故 <math id="M174"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420942&type= 53.25532913 9.99066734 收敛。 3) <math id="M175"><mi>p</mi><mo>≥</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420945&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420944&type= 9.90599918 7.19666672 且 <math id="M176"><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420947&type= 8.55133343 2.79399991 。若 <math id="M177"><mn mathvariant="normal">0</mn><mo><</mo><mi>θ</mi><mi>t</mi><mo><</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420949&type= 16.42533302 2.79399991 ，则 <math id="M178"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420951&type= 52.49333191 9.99066734 发散;若 <math id="M179"><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>θ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420953&type= 16.34066582 3.30199981 ，令 <math id="M180"><mi>N</mi><mo>=</mo><mfenced open="⌊" close="⌋" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420955&type= 20.06599998 8.29733276 ，则 <math id="M181"><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420957&type= 25.23066711 3.55599999 ，故 <math id="M182"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>N</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420959&type= 53.00133133 9.99066734 发散。 综上可知， <math id="M183"><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420961&type= 9.82133293 7.19666672 或 <math id="M184"><mi>θ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420963&type= 8.55133343 2.79399991 是 <math id="M185"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420965&type= 12.95399952 4.65666676 收敛的充要条件。证毕。 <math id="M186"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420967&type= 12.95399952 4.65666676 和 <math id="M187"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420969&type= 12.95399952 4.65666676 的收敛性情况如 表1 表1 。 p 2 <math id="M192"><mi>θ</mi><mo><</mo><mn>1</mn></math> <math id="M193"><mi>θ</mi><mo>></mo><mn>1</mn></math> <math id="M194"><mi>p</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> <math id="M195"><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> 收敛 <math id="M196"><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> 收敛 <math id="M197"><mi>p</mi><mo><</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> <math id="M198"><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> 收敛 <math id="M199"><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> 收敛 <math id="M200"><mi>p</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math> <math id="M201"><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> 收敛 <math id="M202"><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> 收敛 假设条件(d1)、(d2)、(d3)之一成立。接下来我们在 表1 表1 所示条件下证明 <math id="M203"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420981&type= 13.71600056 4.65666676 和 <math id="M204"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420983&type= 13.71600056 4.65666676 是 <math id="M205"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420985&type= 7.45066643 3.55599999 的两个无偏估计量，其中 <math id="M206"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420987&type= 2.62466669 2.62466669 是来自 <math id="M207"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420989&type= 23.62199974 3.89466667 的单个样本。 定理2 若(d1)或 <math id="M208"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420991&type= 16.59466553 7.19666672 成立，则有 <math id="M209"><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext>  </mtext><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420993&type= 45.29666519 4.65666676 ; 若(d2)或 <math id="M210"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420995&type= 16.42533302 7.19666672 成立，则有 <math id="M211"><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420997&type= 44.45000076 4.65666676 。 证 假设(d1)、(d2)、 <math id="M212"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420999&type= 16.42533302 7.19666672 、 <math id="M213"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421001&type= 16.59466553 7.19666672 之一成立， <math id="M214"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421003&type= 2.62466669 2.62466669 是来自 <math id="M215"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421005&type= 23.62199974 3.89466667 的单个样本，则 <math display="block" id="M216"><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>p</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421007&type= 95.50399780 6.77333355 <math display="block" id="M217"><mi>p</mi><msubsup><mo>∫</mo><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mo>∫</mo><mrow><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mo>=</mo><mi>p</mi><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421009&type= 126.49199677 7.28133297 因此， <math display="block" id="M218"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421011&type= 67.90267181 9.05933285 <math display="block" id="M219"><mtable columnalign="left"><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo>⋯</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>k</mi></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mi>N</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421013&type= 145.54200745 63.66933441 当(d1)或 <math id="M220"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421015&type= 16.59466553 7.19666672 成立时， <math id="M221"><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><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421018&type= 49.86866760 9.82133293 ，由于 <math display="block" id="M222"><mfenced open="|" close="|" separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>k</mi></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi><mo>-</mo><mo stretchy="false">|</mo><mi>p</mi><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=3421021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421020&type= 66.12466431 11.26066685 根据控制收敛定理有 <math display="block" id="M223"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><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><mi>E</mi><mfenced separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>k</mi></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>k</mi></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><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=3421023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421022&type= 144.52600098 11.26066685 所以当(d1)或 <math id="M224"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421024&type= 16.59466553 7.19666672 成立时， <math id="M225"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421026&type= 13.71600056 4.65666676 是 <math id="M226"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421028&type= 7.45066643 3.55599999 的一个无偏估计。 在(6)式中用 <math id="M227"><mi>θ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421030&type= 2.87866688 2.70933342 代替 <math id="M228"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421032&type= 1.10066664 2.62466669 ，得到： <math id="M229"><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mi>R</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo>∀</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421034&type= 74.08333588 3.72533321 ，因此 <math display="block" id="M230"><mtable columnalign="left"><mtr><mtd><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>R</mi><mo stretchy="false">(</mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><mi>θ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo>⋯</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mo>-</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mfenced separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mfenced><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mi>N</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421036&type= 163.83000183 88.81533051 当(d2)或 <math id="M231"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421039&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421038&type= 16.42533302 7.19666672 成立时， <math id="M232"><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><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>R</mi><mo stretchy="false">(</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421040&type= 53.76333237 8.12800026 ，由于 <math display="block" id="M233"><mo stretchy="false">|</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo>-</mo><mo stretchy="false">|</mo><mi>p</mi><mo stretchy="false">|</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421042&type= 79.33266449 9.99066734 根据控制收敛定理，有 <math display="block" id="M234"><mtable columnalign="left"><mtr><mtd><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><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><mi>E</mi><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mi>E</mi><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo>></mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421044&type= 94.06466675 21.75933456 所以当(d2)或 <math id="M235"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421046&type= 16.42533302 7.19666672 成立时， <math id="M236"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421048&type= 13.71600056 4.65666676 是 <math id="M237"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421050&type= 7.45066643 3.55599999 的一个无偏估计。证毕。 <math id="M238"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421052&type= 13.71600056 4.65666676 和 <math id="M239"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421054&type= 13.71600056 4.65666676 都是 <math id="M240"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421056&type= 7.45066643 3.55599999 的基于单个样本 <math id="M241"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421058&type= 2.62466669 2.62466669 的无偏估计量，与Nie等 [ 11 11 ] 得到的两个无偏估计形式上一致。他们工作中的 <math id="M242"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421060&type= 2.62466669 2.62466669 服从混合指数分布，这里的 <math id="M243"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421063&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421062&type= 2.62466669 2.62466669 服从混合Gamma分布。 3　 无偏估计的适当性 1 假设条件(d1)、(d2)、(d3)之一成立， <math id="M244"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421064&type= 2.62466669 2.62466669 是来自 <math id="M245"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421066&type= 23.62199974 3.89466667 的单个样本，在 <math id="M246"><mi>p</mi><mo>≠</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421069&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421068&type= 9.73666668 7.19666672 时 <math id="M247"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421070&type= 13.71600056 4.65666676 或 <math id="M248"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421072&type= 13.71600056 4.65666676 是可靠性函数 <math id="M249"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421074&type= 7.45066643 3.55599999 的无偏估计。由于 <math id="M250"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421077&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421076&type= 7.45066643 3.55599999 取值范围为[0,1]，因此要保证 <math id="M251"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421078&type= 29.71799850 4.65666676 的取值范围也为[0,1]，称此时的估计是适当的。接下来讨论在什么样的条件下 <math id="M252"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421080&type= 29.71799850 4.65666676 才是适当的。 定理3 若(d1)成立，则 <math id="M253"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421082&type= 13.71600056 4.65666676 是 <math id="M254"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421084&type= 7.45066643 3.55599999 的一个适当的无偏估计;若(d2)成立，则 <math id="M255"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421086&type= 13.71600056 4.65666676 是 <math id="M256"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421088&type= 7.45066643 3.55599999 的一个适当的无偏估计。 证 若(d1)成立，根据 表1 表1 ，此时 <math id="M257"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421090&type= 12.95399952 4.65666676 收敛。由于 <math id="M258"><mn mathvariant="normal">0</mn><mo><</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421092&type= 23.36800003 8.04333305 ，因此 <math id="M259"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421094&type= 12.95399952 4.65666676 满足： <math id="M260"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo><</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421096&type= 52.49333191 9.82133293 。从而 <math id="M261"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421098&type= 13.71600056 4.65666676 是 <math id="M262"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421100&type= 7.45066643 3.55599999 适当的无偏估计。 若(d2)成立，根据 表1 表1 ，此时 <math id="M263"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421102&type= 12.95399952 4.65666676 收敛。由于 <math id="M264"><mn mathvariant="normal">0</mn><mo><</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421104&type= 29.80266571 8.12800026 ，因此 <math id="M265"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421106&type= 12.95399952 4.65666676 满足： <math id="M266"><mn mathvariant="normal">0</mn><mo>≤</mo><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo><</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421108&type= 65.36266327 9.99066734 。从而 <math id="M267"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421110&type= 13.71600056 4.65666676 是 <math id="M268"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421112&type= 7.45066643 3.55599999 适当的无偏估计。证毕。 由 表1 表1 可知，若(d1)成立， <math id="M269"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421114&type= 12.95399952 4.65666676 不是收敛的。若(d2)成立， <math id="M270"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421116&type= 12.95399952 4.65666676 不是收敛的。 当 <math id="M271"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421118&type= 16.59466553 7.19666672 时， <math id="M272"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421120&type= 13.71600056 4.65666676 是 <math id="M273"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421122&type= 7.45066643 3.55599999 的无偏估计，但不是适当的;当 <math id="M274"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421124&type= 16.42533302 7.19666672 时 <math id="M275"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421126&type= 13.71600056 4.65666676 是 <math id="M276"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421128&type= 7.45066643 3.55599999 的无偏估计，但不是适当的。下面说明之。 1) 对于 <math id="M277"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421130&type= 12.95399952 4.65666676 ，根据定理2，当 <math id="M278"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421132&type= 16.59466553 7.19666672 时 <math id="M279"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421134&type= 13.71600056 4.65666676 是 <math id="M280"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421136&type= 7.45066643 3.55599999 的一个无偏估计。根据 <math id="M281"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421138&type= 1.77800000 2.70933342 的范围分两种情况讨论 <math id="M282"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421141&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421140&type= 13.71600056 4.65666676 的适当性。 a) <math id="M283"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421142&type= 26.92399788 7.19666672 。若 <math id="M284"><mn mathvariant="normal">0</mn><mo><</mo><mi>t</mi><mo><</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421144&type= 14.64733315 2.79399991 ，令 <math id="M285"><mi>N</mi><mo>=</mo><mfenced open="⌈" close="⌉" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>t</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421146&type= 27.09333420 8.29733276 ，则 <math id="M286"><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>N</mi></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421148&type= 30.64933395 3.55599999 ，故 <math display="block" id="M287"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421150&type= 68.41066742 9.82133293 由于 <math id="M288"><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421152&type= 16.76399994 8.04333305 ，因此当 <math id="M289"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421154&type= 2.87866688 2.70933342 为偶数时 <math id="M290"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421156&type= 12.95399952 4.65666676 大于1。所以 <math id="M291"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421158&type= 13.71600056 4.65666676 不是适当的。 b) <math id="M292"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421160&type= 26.92399788 7.19666672 。若 <math id="M293"><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421162&type= 14.56266594 3.30199981 ，令 <math id="M294"><mi>N</mi><mo>=</mo><mfenced open="⌊" close="⌋" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>t</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421164&type= 20.06599998 8.29733276 ，则 <math id="M295"><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mi>N</mi></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421166&type= 30.64933395 3.55599999 ，故 <math display="block" id="M296"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421168&type= 66.46333313 9.82133293 由于 <math id="M297"><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421170&type= 16.76399994 8.04333305 ，因此当 <math id="M298"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421172&type= 2.87866688 2.70933342 为偶数时 <math id="M299"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421174&type= 12.95399952 4.65666676 小于0。所以 <math id="M300"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421176&type= 13.71600056 4.65666676 不是适当的。 2) 对于 <math id="M301"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421178&type= 12.95399952 4.65666676 ，根据定理2，当 <math id="M302"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421180&type= 16.42533302 7.19666672 时 <math id="M303"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421182&type= 13.71600056 4.65666676 是 <math id="M304"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421184&type= 7.45066643 3.55599999 的一个无偏估计。根据 <math id="M305"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421186&type= 1.77800000 2.70933342 的范围分两种情况讨论 <math id="M306"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421188&type= 13.71600056 4.65666676 的适当性。 a) <math id="M307"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421190&type= 27.26266861 7.19666672 。若 <math id="M308"><mn mathvariant="normal">0</mn><mo><</mo><mi>x</mi><mo>≤</mo><mi>θ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421192&type= 16.34066582 3.30199981 ，令 <math id="M309"><mi>N</mi><mo>=</mo><mfenced open="⌊" close="⌋" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421194&type= 20.06599998 8.29733276 ，则 <math id="M310"><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421196&type= 25.23066711 3.55599999 ，故 <math display="block" id="M311"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>N</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421198&type= 78.31666565 9.99066734 由于 <math id="M312"><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421200&type= 23.19866753 8.12800026 ，因此当 <math id="M313"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421202&type= 2.87866688 2.70933342 为奇数时 <math id="M314"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421204&type= 12.95399952 4.65666676 小于0。所以 <math id="M315"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421206&type= 13.71600056 4.65666676 不是适当的。 b) <math id="M316"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421208&type= 27.26266861 7.19666672 。若 <math id="M317"><mn mathvariant="normal">0</mn><mo><</mo><mi>θ</mi><mi>t</mi><mo><</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421210&type= 16.42533302 2.79399991 ，令 <math id="M318"><mi>N</mi><mo>=</mo><mfenced open="⌈" close="⌉" separators="|"><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfrac><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421212&type= 27.09333420 8.29733276 ，则 <math id="M319"><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi></mrow></msup><mi>t</mi><mo><</mo><mi>x</mi><mo>≤</mo><msup><mrow><mi>θ</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421214&type= 25.23066711 3.55599999 ，故 <math display="block" id="M320"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></munderover><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mi>N</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421216&type= 84.75132751 9.99066734 由于 <math id="M321"><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421218&type= 23.19866753 8.12800026 ，因此当 <math id="M322"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421220&type= 2.87866688 2.70933342 为奇数时 <math id="M323"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421222&type= 12.95399952 4.65666676 大于1。所以 <math id="M324"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421224&type= 13.71600056 4.65666676 不是适当的。 根据上面的讨论， <math id="M325"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421226&type= 13.71600056 4.65666676 和 <math id="M326"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421228&type= 13.71600056 4.65666676 不可能同时是 <math id="M327"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421230&type= 7.45066643 3.55599999 的适当的无偏估计。可以发现，只有当混合权重 <math id="M328"><mo stretchy="false">(</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421232&type= 3.89466667 3.72533321 或 <math id="M329"><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421234&type= 9.90599918 3.72533321 其中一个为负值的时候， <math id="M330"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421236&type= 13.71600056 4.65666676 或 <math id="M331"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421238&type= 13.71600056 4.65666676 才可能是 <math id="M332"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421240&type= 7.45066643 3.55599999 的适当的无偏估计。这个结论和Nie等 [ 11 11 ] 指数分布情形下的结论一致。 4　 数值模拟 1 假设条件(d1)成立， <math id="M333"><mi>T</mi><mo>~</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421242&type= 19.55799866 3.89466667 ，其概率密度函数为 <math id="M334"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421244&type= 16.00200081 3.89466667 ，其可靠性函数为 <math id="M335"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421246&type= 37.50733566 6.77333355 。混合Gamma分布记为 <math id="M336"><mi>p</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="normal">G</mi><mi mathvariant="normal">a</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><msup><mrow><mi>θ</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421248&type= 50.79999924 4.14866638 ，其概率密度函数为 <math id="M337"><mi>ψ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421250&type= 23.62199974 3.89466667 ，其分布函数为 <math id="M338"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>x</mi></mrow></msubsup><mrow><mi>ψ</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421252&type= 45.88933182 6.77333355 。用逆采样方法生成混合Gamma分布的随机观测:首先，假设随机变量 <math id="M339"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421254&type= 3.04799986 2.62466669 服从区间[0,1]上的均匀分布，利用计算机生成 <math id="M340"><mi>U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421256&type= 3.04799986 2.62466669 的一个随机观测 <math id="M341"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421258&type= 1.86266661 2.62466669 ，然后求解 <math id="M342"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421260&type= 2.03200006 2.62466669 使得 <math id="M343"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421262&type= 14.98599911 3.55599999 。取 <math id="M344"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421264&type= 2.03200006 2.62466669 为混合Gamma分布的一个随机观测。根据 <math id="M345"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421266&type= 2.03200006 2.62466669 得到 <math id="M346"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421268&type= 18.96533394 3.55599999 的无偏估计 <math id="M347"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421270&type= 12.95399952 4.65666676 ，并与真实的可靠性函数 <math id="M348"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421272&type= 7.45066643 3.55599999 进行比较。 取 <math id="M349"><mi>p</mi><mo>=</mo><mn mathvariant="normal">1.1</mn><mo>,</mo><mi>θ</mi><mo>=</mo><mn mathvariant="normal">0.8</mn><mo>,</mo><mi>α</mi><mo>=</mo><mn mathvariant="normal">4</mn><mo>,</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421274&type= 45.63533401 3.81000018 ，当用计算机生成单个随机观测 <math id="M350"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421276&type= 2.03200006 2.62466669 后，根据(4)式求出 <math id="M351"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421278&type= 12.95399952 4.65666676 ，并与真实的可靠性函数 <math id="M352"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421280&type= 7.45066643 3.55599999 作比较。当生成不同的单个随机观测时， <math id="M353"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421282&type= 12.95399952 4.65666676 和 <math id="M354"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421284&type= 7.45066643 3.55599999 的比较如 图1 图1 和 图2 图2 。 图1 图1 为 <math id="M367"><mi>x</mi><mo>=</mo><mn mathvariant="normal">3.2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421316&type= 11.68400002 2.70933342 时的模拟结果， 图2 图2 为 <math id="M368"><mi>x</mi><mo>=</mo><mn mathvariant="normal">5.4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421318&type= 11.68400002 2.79399991 时的模拟结果。可以看出， 图1 图1 和 图2 图2 中 <math id="M369"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421320&type= 12.95399952 4.65666676 能够较好地估计 <math id="M370"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421322&type= 7.45066643 3.55599999 。 <math id="M371"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421324&type= 12.95399952 4.65666676 的估计效果取决于混合Gamma分布的随机观测 <math id="M372"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421326&type= 2.03200006 2.62466669 。若在单个观测的情况下利用经验分布函数去估计 <math id="M373"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421328&type= 7.45066643 3.55599999 ，只能得到只有一个阶梯的阶梯函数(取值为0或1)。由(4)式知 <math id="M374"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421330&type= 12.95399952 4.65666676 是一个阶梯函数，且有好几个阶梯，因此在单个观测的情况下， <math id="M375"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421332&type= 12.95399952 4.65666676 能够比经验分布函数更好地估计 <math id="M376"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421334&type= 7.45066643 3.55599999 。所以，本文的工作是有意义的。 接下来验证 <math id="M377"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421336&type= 12.95399952 4.65666676 的无偏性。每进行一次试验，得到一个观测，进而可得一个无偏估计 <math id="M378"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421338&type= 12.95399952 4.65666676 。 当进行大量试验时，希望由此得到的大量 <math id="M379"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421340&type= 12.95399952 4.65666676 的平均值能精确地估计 <math id="M380"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421342&type= 7.45066643 3.55599999 。取 <math id="M381"><mi>p</mi><mo>=</mo><mn mathvariant="normal">1.1</mn><mo>,</mo><mi>θ</mi><mo>=</mo><mn mathvariant="normal">0.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421344&type= 24.72266769 3.55599999 ，重复进行500次随机试验，一共生成500个观测 <math id="M382"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mtext> </mtext><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">500</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421346&type= 29.12533188 4.06400013 ，其中 <math id="M383"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421348&type= 2.79399991 3.55599999 为第 <math id="M384"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421350&type= 1.10066664 2.62466669 次试验得到的观测。计算 <math id="M385"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421352&type= 7.45066643 3.55599999 的500个无偏估计 <math id="M386"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">500</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421354&type= 40.47066879 4.65666676 ，其平均值记为 <math display="block" id="M387"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">500</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">500</mn></mrow></munderover><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421356&type= 38.69266510 8.29733276 当 <math id="M388"><mi>α</mi><mo>,</mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421358&type= 5.67266655 3.72533321 取不同的值时， <math id="M389"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></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=3421361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421360&type= 8.12800026 4.65666676 和 <math id="M390"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421362&type= 7.45066643 3.55599999 的比较如 图3 图3 和 图4 图4 。 图3 图3 为 <math id="M403"><mi>α</mi><mo>=</mo><mn mathvariant="normal">4</mn><mo>,</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421394&type= 19.13466644 3.81000018 时的模拟结果， 图4 图4 为 <math id="M404"><mi>α</mi><mo>=</mo><mn mathvariant="normal">3</mn><mo>,</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421396&type= 19.13466644 3.72533321 时的模拟结果。 图3 图3 与 图1 图1 和2均是在 <math id="M405"><mi>p</mi><mo>=</mo><mn mathvariant="normal">1.1</mn><mo>,</mo><mi>θ</mi><mo>=</mo><mn mathvariant="normal">0.8</mn><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421398&type= 26.58533287 3.55599999 <math id="M406"><mi>α</mi><mo>=</mo><mn mathvariant="normal">4</mn><mo>,</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421400&type= 19.13466644 3.81000018 时的模拟结果，不同的是 图1 图1 和 图2 图2 是根据单个无偏估计的模拟，而 图3 图3 是根据500个无偏估计的模拟。通过 图3 图3 与 图1 图1 和2的比较可以看出:当用到的无偏估计的数量很多时，多个估计的平均值能比较精确的拟合 <math id="M407"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421402&type= 7.45066643 3.55599999 ，即验证了 <math id="M408"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421404&type= 12.95399952 4.65666676 的无偏性。 图4 图4 是 <math id="M409"><mi>α</mi><mo>=</mo><mn mathvariant="normal">3</mn><mo>,</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421406&type= 19.13466644 3.72533321 时利用500个无偏估计的模拟结果，进一步验证了 <math id="M410"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421408&type= 12.95399952 4.65666676 的无偏性。 5　 结 语 1 本文基于混合Gamma分布的单个观测值讨论了Gamma分布可靠性函数的无偏估计及其相关性质。 首先，给出了混合Gamma分布的概率密度函数可定义的充要条件。其次，在 <math id="M411"><mi>p</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>θ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421410&type= 18.37266731 3.64066648 已知， <math id="M412"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421412&type= 19.13466644 3.81000018 未知的情况下，基于混合Gamma分布的单个观测值给出了Gamma分布的可靠性函数 <math id="M413"><mi>R</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421414&type= 7.45066643 3.55599999 的无偏估计 <math id="M414"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421416&type= 12.95399952 4.65666676 和 <math id="M415"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421418&type= 12.95399952 4.65666676 。接下来，我们讨论了 <math id="M416"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421420&type= 12.95399952 4.65666676 和 <math id="M417"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421422&type= 12.95399952 4.65666676 的适当性，发现当 <math id="M418"><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421425&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421424&type= 8.55133343 3.64066648 且 <math id="M419"><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot><mo>≤</mo><mi>θ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421426&type= 27.17800140 9.31333351 时 <math id="M420"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421428&type= 12.95399952 4.65666676 在[0,1]之间，当 <math id="M421"><mi>p</mi><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421430&type= 8.55133343 3.64066648 且 <math id="M422"><mn mathvariant="normal">1</mn><mo><</mo><mi>θ</mi><mo>≤</mo><mroot><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421432&type= 26.75466537 9.31333351 时 <math id="M423"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421434&type= 12.95399952 4.65666676 在[0,1]之间。最后，我们利用计算机生成模拟数据，验证了 <math id="M424"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421436&type= 12.95399952 4.65666676 和 <math id="M425"><msub><mrow><mi>g</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3421438&type= 12.95399952 4.65666676 的无偏性。 Nie等 [ 11 11 ] 在混合指数分布的情况给出了指数分布可靠性函数的无偏估计及其相关性质，本文中的相关结论是基于混合Gamma分布得到的，是其工作的推广。 参考文献 NEWCOMB S . 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Unbiased estimation of reliability function from a mixture of two exponential distributions based on a single observation 黄卓 , 李苏军 , 郭波 . 基于混合Gamma分布的通用可靠性寿命数据拟合方法 [J]. 航空学报 , 2008 , 29 ( 2 )： 379 - 386 . 基于混合Gamma分布的通用可靠性寿命数据拟合方法 HUANG Z , LI S J , GUO B . Fitting method for general reliability life data based on mixed Gamma distribution [J]. Journal of Aeronautics , 2008 , 29 ( 2 )： 379 - 386 (Ch) . Fitting method for general reliability life data based on mixed Gamma distribution 10.14188/j.1671-8836.2019.0102 O212 A 1671-8836（2020）03-0205-10 WANG Borui,LIU Luqin.Unbiased Estimation of Reliability Function Based on One Single Observation from a Mixture of Gamma Distributions [J]. J Wuhan Univ （ Nat Sci Ed ）,2020,66(3):205-214. DOI: 10.14188/j.1671-8836.2019.0102 http://dx.doi.org/10.14188/j.1671-8836.2019.0102 (Ch). WANG Borui,LIU Luqin.Unbiased Estimation of Reliability Function Based on One Single Observation from a Mixture of Gamma Distributions [J].J Wuhan Univ（Nat Sci Ed）,2020,66(3):205-214. DOI:10.14188/j.1671-8836.2019.0102(Ch). 王博锐,刘禄勤.基于混合Gamma分布单个观测的可靠性函数无偏估计[J].武汉大学学报（理学版）,2020,66(3):205-214. DOI:10.14188/j.1671-8836.2019.0102 王博锐,刘禄勤.基于混合Gamma分布单个观测的可靠性函数无偏估计[J].武汉大学学报（理学版）,2020,66(3):205-214. DOI:10.14188/j.1671-8836.2019.0102 国家自然科学基金(11171263) 2019-05-03 2020-06-24 2020-09-30 武汉大学学报（理学版） 3 66