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威布尔⁃离散位相型分布
数学 | 更新时间：2020-11-28
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威布尔⁃离散位相型分布
- Weibull⁃Discrete Phase⁃Type Distribution
- 武汉大学学报（理学版）
- 作者机构：
1.武汉大学 数学与统计学院，湖北 武汉 430072
- 作者简介：
张志方，男，硕士生，主要研究数理统计方向。 E-mail：1064649422@qq.com
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- 基金信息：国家自然科学基金(1171263)
- DOI：10.14188/j.1671-8836.2018.0108 中图分类号： O211.3
纸质出版日期：2020-06-24，
收稿日期：2018-04-15，
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引用本文
张志方,刘禄勤.威布尔⁃离散位相型分布[J].武汉大学学报（理学版）,2020,66(3):215-223.
ZHANG Zhifang,LIU Luqin.Weibull-Discrete Phase-Type Distribution [J].J Wuhan Univ （Nat Sci Ed）,2020,66(3):215-223.
张志方,刘禄勤.威布尔⁃离散位相型分布[J].武汉大学学报（理学版）,2020,66(3):215-223. DOI：10.14188/j.1671-8836.2018.0108
ZHANG Zhifang,LIU Luqin.Weibull-Discrete Phase-Type Distribution [J].J Wuhan Univ （Nat Sci Ed）,2020,66(3):215-223. DOI：10.14188/j.1671-8836.2018.0108(Ch).
威布尔⁃离散位相型分布
威布尔⁃离散位相型分布 Weibull⁃Discrete Phase⁃Type Distribution 1 first-author 张志方，男，硕士生，主要研究数理统计方向。 E-mail：1064649422@qq.com 张志方，男，硕士生，主要研究数理统计方向。 E-mail：1064649422@qq.com 1064649422@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 提出了一类具有多种形状危险率的新型寿命分布，即由威布尔分布和离散位相型分布混合生成的威布尔-离散位相型分布，给出了它的密度函数、危险率函数和 <math id="M1"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419501&type= 1.43933344 2.37066674 阶矩；研究了威布尔-2阶几何分布的相关性质，证明了其参数极大似然估计的相合性和渐近正态性。数值模拟与实例研究结果表明：威布尔-2阶几何分布具有良好的有限样本性质及具有一定的实用性。 In this paper，a new class of lifetime distributions with various shapes，named Weibull-discrete phase-type distribution which is obtained by compounding Weibull distribution and discrete phase-type distribution is introduced. The density，failure rate function，and the k⁃ th moment of this distribution are provided. Various properties of Weibull-geometric distribution of order 2 are researched and the consistency and asymptotic normality of its maximum likelihood estimation of parameters are proved. Simulation study and illustrating examples based on real data are presented. 威布尔分布 离散位相型分布 混合分布 2阶几何分布 极大似然估计 Weibull distribution discrete phase-type distribution mixed distribution geometric distribution of order 2 maximum likelihood estimation 0 引 言 1 寿命分布是统计分布的重要组成部分，是可靠性研究和可靠性工程应用的基础。随着寿命分布理论研究的深入和科学技术的发展，现有的寿命分布模型已不能满足人们的需要。为更好地处理寿命数据，人们便提出了多种产生寿命分布的机制。本文将威布尔分布与离散位相型分布“混合”,形成了一类新的寿命分布。这类分布起源于随机最小值分布。 设 <math id="M2"><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419503&type= 15.91733360 3.97933340 为独立同分布随机变量序列，其分布函数为 <math id="M3"><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419505&type= 8.80533314 3.55599999 ； <math id="M4"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419507&type= 1.60866666 3.55599999 为与 <math id="M5"><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419509&type= 15.91733360 3.97933340 独立的取正整数值的随机变量，其分布为 <math id="M6"><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419513&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419512&type= 9.14400005 3.55599999 ；称 <math id="M7"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419514&type= 13.71600056 4.48733330 <math id="M8"><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419516&type= 6.01133347 3.97933340 <math id="M9"><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>ξ</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419520&type= 10.92199993 4.48733330 服从的分布为G-H分布。Adamidis和Loukas [ 1 1 ] 通过这种方式将指数分布与几何分布混合得到了一种具有递减危险率的寿命分布：指数-几何分布(EG分布)。跟随他们的思路，Kus [ 2 2 ] 提出了指数-泊松分布(EP分布)，Tahmasbi和Rezaei [ 3 3 ] 提出了指数-对数分布(EL分布)，Hajebi [ 4 4 ] 提出了指数-负二项分布(ENB分布)，Eryilmaz [ 5 5 ] 通过混合指数分布与离散位相型分布 [ 6 6 ] (discrete phase-type distribution)提出了指数-离散位相型分布(EDPH分布)。上述混合分布都是基于指数分布，并且这些分布都有递减危险率。 在实际应用中，由于威布尔分布的危险率函数形状多样，故也常被用来做生存分析。用威布尔分布代替指数分布进行混合是一种更好的想法。Ciumara和Preda [ 7 7 ] 提出了威布尔-对数分布(WL分布)，Barreto-Souza [ 8 8 ] 提出了威布尔-几何分布(WG分布)，Rodrigues [ 9 9 ] 提出了威布尔-负二项分布(WNB分布)，Lu和Shi [ 10 10 ] 提出了威布尔-泊松分布(WP分布)。 离散位相型分布是几何分布推广，其在位相意义下也具有无记忆性。离散位相型分布是一个有限状态马氏链寿命的分布，它的每一个暂留态称为位相，一旦到达某位相，在其上的生存时间服从几何分布。Neuts [ 6 6 ] 给出了离散位相型分布的定义和一些性质及其在排队理论中的应用。文献[ 11 11 , 12 12 , 13 13 , 14 14 ]进一步介绍了离散位相型分布的一些性质及应用。本文通过选取 <math id="M10"><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419522&type= 15.91733360 3.97933340 服从威布尔分布和 <math id="M11"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419524&type= 1.60866666 3.55599999 服从离散位相型分布提出了威布尔-离散位相型分布(WDPH分布)，推广了文献[ 5 5 ]的相关工作。 本文结构如下：第1节定义了威布尔-离散位相型分布，并给出了其密度函数、危险率函数和 <math id="M12"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419528&type= 1.69333339 2.79399991 阶矩；第2节研究了一类特殊的威布尔-离散位相型分布：威布尔-2阶几何分布( <math id="M13"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419530&type= 7.62000036 3.72533321 分布)，给出了其密度函数、危险率函数的一些性质以及其参数的极大似然估计的存在性、相合性和渐近正态性；第3节对威布尔-2阶几何分布进行了数值模拟和实例研究，展现了 <math id="M14"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419532&type= 7.62000036 3.72533321 分布的实用性。 1 威布尔-离散位相型分布的定义及性质 1 设 <math id="M15"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419534&type= 23.19866753 4.48733330 ，其中 <math id="M16"><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419536&type= 9.56733322 4.14866638 且 <math id="M17"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>d</mi></mrow></munderover><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419538&type= 14.64733315 8.38199997 ； <math id="M18"><mi mathvariant="bold-italic">Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419540&type= 13.46199894 4.65666676 为 <math id="M19"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419542&type= 2.20133328 2.79399991 阶矩阵， <math id="M20"><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn></math> 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mathvariant="normal">1</mn></mrow><mrow><mi>d</mi></mrow></munderover><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>k</mi><mi>j</mi></mrow></msub></mrow><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419550&type= 15.57866669 8.80533314 ，对 <math id="M24"><mo>∀</mo><mn mathvariant="normal">1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419552&type= 16.76399994 3.30199981 都存在 <math id="M25"><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419554&type= 15.15533447 4.40266657 满足 <math id="M26"><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo>⋯</mo><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>j</mi></mrow><mrow><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mi>k</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419556&type= 26.83933258 4.06400013 。记 <math id="M27"><mi mathvariant="bold-italic">t</mi><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419558&type= 20.82799911 3.55599999 ，其中 <math id="M28"><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419560&type= 1.43933344 2.62466669 为 <math id="M29"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419562&type= 2.20133328 2.79399991 阶单位矩阵， <math id="M30"><mi mathvariant="bold-italic">e</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419564&type= 21.92866707 3.81000018 。令 <math display="block" id="M31"><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">a</mi><msup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419566&type= 18.96533394 4.65666676 , <math display="block" id="M32"><mi>n</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419568&type= 17.94933319 3.13266659 易证 <math id="M33"><mo stretchy="false">{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419570&type= 17.44133377 4.74133348 为一概率分布。 定义1 称(1)式确定的概率分布 <math id="M34"><mo stretchy="false">{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419572&type= 17.44133377 4.74133348 是以 <math id="M35"><mi mathvariant="bold-italic">a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419574&type= 1.94733334 2.62466669 和 <math id="M36"><mi mathvariant="bold-italic">Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419576&type= 2.96333337 3.04799986 为参数的 <math id="M37"><mo stretchy="false">(</mo><mi>d</mi><mi mathvariant="normal">阶</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419578&type= 9.48266697 3.72533321 离散位相型分布，记为 <math id="M38"><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419581&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419580&type= 18.20333290 3.55599999 。 若随机变量 <math id="M39"><mi>ξ</mi><mo>∼</mo><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419582&type= 24.21466637 3.72533321 ，则可以构建一状态空间为 <math id="M40"><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419584&type= 29.37933350 3.55599999 ，初始分布为 <math id="M41"><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419586&type= 8.72066689 3.55599999 ，转移概率矩阵为 <math id="M42"><mi mathvariant="bold-italic">P</mi><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">Q</mi></mtd><mtd><mi mathvariant="bold-italic">t</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419588&type= 32.25799942 9.73666668 的时齐马氏链 <math id="M43"><mo stretchy="false">{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>=</mo><mn mathvariant="normal">0,1</mn><mo>,</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419590&type= 30.98800087 4.48733330 ，使得 <math id="M44"><mi>ξ</mi><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">{</mo><mi>n</mi><mo>:</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>d</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419592&type= 38.18466568 4.65666676 ，且状态 <math id="M45"><mi>d</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419594&type= 8.97466660 2.96333337 是吸收态，状态 <math id="M46"><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>，</mo><mi>d</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419596&type= 23.79133415 3.55599999 是暂留态。 离散位相型分布是几何分布的推广，同时它也包括其他一些常用的分布如：混合几何分布、负二项分布，混合负二项分布等。离散位相型分布一方面能够把一些常用分布用矩阵代数的方法表示出来，形式简洁统一，另一方面选取不同的 <math id="M47"><mi mathvariant="bold-italic">a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419598&type= 1.94733334 2.62466669 和 <math id="M48"><mi mathvariant="bold-italic">Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419600&type= 2.96333337 3.04799986 时就能得到不同的离散分布，分布包含的范围得到很大地拓展。 威布尔分布 <math id="M49"><mi mathvariant="normal">W</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=3419603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419602&type= 12.95399952 3.89466667 是指数分布的推广，其密度函数为 <math id="M50"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>,</mo><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419604&type= 39.96266937 4.65666676 ， <math id="M51"><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419606&type= 8.29733276 2.79399991 为尺度参数， <math id="M52"><mi>β</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419608&type= 8.38199997 3.81000018 为形状参数，其分布函数为 <math id="M53"><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419610&type= 25.90799904 3.97933340 。 定义2 设独立同分布 <math id="M54"><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419612&type= 4.06400013 3.55599999 <math id="M55"><mo>∼</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419614&type= 3.13266659 2.62466669 <math id="M56"><mi mathvariant="normal">W</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=3419617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419616&type= 12.95399952 3.89466667 <math id="M57"><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419618&type= 1.86266661 3.04799986 <math id="M58"><mi>ξ</mi><mo>~</mo><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419620&type= 18.54199982 3.72533321 <math id="M59"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419622&type= 4.48733330 3.55599999 且与 <math id="M60"><mo stretchy="false">{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">1</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419624&type= 17.69533348 4.57200003 独立，称 <math id="M61"><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>ξ</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419626&type= 30.22600174 4.48733330 服从的分布为威布尔-离散位相型分布，记为 <math id="M62"><mi mathvariant="normal">W</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419628&type= 29.29466629 3.89466667 。 定理1 若 <math id="M63"><mi>X</mi><mo>∼</mo><mi mathvariant="normal">W</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419630&type= 36.40666580 3.89466667 ，则其分布函数、密度函数和危险率函数分别为 <math display="block" id="M64"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">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=3419633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419632&type= 63.66933441 4.23333359 <math display="block" id="M65"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">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=3419635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419634&type= 67.31000519 4.65666676 <math display="block" id="M66"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mfrac><mrow><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow></mfrac><mo>,</mo><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419636&type= 61.72200012 9.22866726 证 <math id="M67"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419638&type= 2.62466669 2.62466669 的生存函数为 <math display="block" id="M68"><mover><mrow><mi>F</mi></mrow><mo stretchy="true">¯</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419640&type= 20.48933411 3.81000018 <math display="block" id="M69"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419642&type= 12.53066635 3.55599999 <math display="block" id="M70"><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>ξ</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419644&type= 43.43400192 4.48733330 = <math display="block" id="M71"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419646&type= 16.93333435 4.48733330 <math display="block" id="M72"><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419648&type= 18.37266731 3.72533321 <math display="block" id="M73"><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419650&type= 31.83466721 4.48733330 <math display="block" id="M74"><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419652&type= 22.94466591 3.72533321 = <math display="block" id="M75"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>G</mi><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>P</mi></mrow><mo stretchy="false">(</mo><mi>ξ</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419654&type= 39.96266937 8.29733276 <math display="block" id="M76"><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>ξ</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419656&type= 20.40466690 4.23333359 上式即为随机变量 <math id="M77"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419658&type= 1.60866666 3.55599999 的概率母函数在 <math id="M78"><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419660&type= 6.68866634 3.38666677 点的值。 <math id="M79"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419662&type= 1.60866666 3.55599999 的概率母函数 <math id="M80"><msub><mrow><mi>Φ</mi></mrow><mrow><mi>ξ</mi></mrow></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>z</mi></mrow><mrow><mi>ξ</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mi>z</mi><mi mathvariant="bold-italic">a</mi><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi>z</mi><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419664&type= 49.10666656 4.65666676 , <math id="M81"><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419666&type= 8.97466660 3.64066648 ，于是 <math id="M82"><mover><mrow><mi>F</mi></mrow><mo stretchy="true">¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419668&type= 45.97399902 3.97933340 <math id="M83"><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419670&type= 1.18533337 2.62466669 。遂得(2)式。 由(2)式易得(3)式和(4)式。证毕。 威布尔-离散位相型分布即 <math id="M84"><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419672&type= 8.80533314 3.55599999 为威布尔分布， <math id="M85"><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419674&type= 9.14400005 3.55599999 为离散位相型分布的G-H分布。易验证指数-几何分布，指数-负二项分布，威布尔-几何分布，威布尔-负二项分布均为威布尔-离散位相型分布特例。 定理 2 若 <math id="M86"><mi>X</mi><mo>∼</mo><mi mathvariant="normal">W</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419676&type= 36.40666580 3.89466667 ，则 <math id="M87"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419678&type= 2.62466669 2.62466669 的 <math id="M88"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419680&type= 1.69333339 2.79399991 阶矩为 <math display="block" id="M89"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>α</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>ξ</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419682&type= 48.51399994 8.72066689 特别地，当 <math id="M90"><mi>k</mi><mo>=</mo><mi>n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419685&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419684&type= 10.32933331 3.81000018 时， <math id="M91"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mi>β</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><msup><mrow><mi>α</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>ξ</mi></mrow><mrow><mo>-</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419686&type= 33.52799988 7.45066643 。 证 由定理1的证明知 <math display="block" id="M92"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>n</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419688&type= 47.75199890 8.29733276 ， <math display="block" id="M93"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mi>n</mi><mi>α</mi><mi>β</mi><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>n</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419690&type= 67.56400299 8.29733276 ， <math display="block" id="M94"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mi>n</mi><mi>α</mi><mi>β</mi><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>=</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>n</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419693&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419692&type= 67.39466858 8.29733276 计算知 <math id="M95"><mi>n</mi><mi>α</mi><mi>β</mi><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>n</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>n</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419694&type= 64.93933105 8.72066689 ，于是 <math id="M96"><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>α</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><mi>ξ</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419696&type= 48.51399994 8.72066689 。证毕。 定理3 若 <math id="M97"><mi>X</mi><mo>∼</mo><mi mathvariant="normal">W</mi><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419698&type= 36.40666580 3.89466667 ， <math id="M98"><mi>ξ</mi><mo>∼</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419700&type= 5.41866684 3.55599999 <math id="M99"><mi mathvariant="normal">D</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">H</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">a</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419702&type= 13.71600056 3.55599999 <math id="M100"><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419704&type= 4.48733330 3.55599999 ，则有 1） <math id="M101"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419706&type= 2.62466669 2.62466669 关于 <math id="M102"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419708&type= 1.60866666 3.55599999 的条件期望为 <math display="block" id="M103"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">|</mo><mi>ξ</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>β</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419710&type= 49.02199936 8.72066689 2） <math id="M104"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419712&type= 1.60866666 3.55599999 关于 <math id="M105"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419714&type= 2.62466669 2.62466669 的条件期望为 <math display="block" id="M106"><mi>E</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419716&type= 59.85932922 9.22866726 证 1） 当 <math id="M107"><mi>ξ</mi><mo>=</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419719&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419718&type= 8.21266651 3.64066648 时， <math id="M108"><mi>X</mi><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>k</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419720&type= 37.93066788 4.48733330 ， <math id="M109"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419722&type= 2.62466669 2.62466669 关于 <math id="M110"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419725&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419724&type= 1.60866666 3.55599999 的条件概率密度函数为 <math display="block" id="M111"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">|</mo><mi>ξ</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>G</mi><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>k</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419726&type= 79.33266449 4.65666676 此即为形状参数为 <math id="M112"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419728&type= 1.94733334 3.72533321 ，尺度参数为 <math id="M113"><mi>k</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419730&type= 3.55599999 2.79399991 的威布尔分布，于是 <math id="M114"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">|</mo><mi>ξ</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>k</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>β</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419732&type= 49.02199936 8.72066689 。 2) <math id="M115"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419734&type= 2.62466669 2.62466669 关于 <math id="M116"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419736&type= 1.60866666 3.55599999 的条件概率密度函数为 <math id="M117"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">|</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>G</mi><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>n</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419738&type= 73.40600586 4.65666676 则 <math id="M118"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419740&type= 2.62466669 2.62466669 和 <math id="M119"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419742&type= 1.60866666 3.55599999 的联合概率密度函数为 <math display="block" id="M120"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">|</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>n</mi><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>n</mi><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="bold-italic">a</mi><msup><mrow><mi mathvariant="bold-italic">Q</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419745&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419744&type= 67.56400299 5.58799982 由(3)和(6)式知， <math display="block" id="M121"><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>n</mi><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419746&type= 61.38333511 9.22866726 在给定 <math id="M122"><mi>X</mi><mo>=</mo><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419748&type= 9.73666668 2.62466669 条件下 <math id="M123"><mi>ξ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419750&type= 1.60866666 3.55599999 的条件期望为 <math display="block" id="M124"><mi>E</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419752&type= 22.43666649 3.72533321 <math display="block" id="M125"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419755&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419754&type= 57.57333374 8.63599968 通过计算得 <math display="block" id="M126"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419756&type= 50.96933365 8.29733276 <math display="block" id="M127"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mn mathvariant="normal">2</mn><mi>n</mi></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419758&type= 51.13866806 8.29733276 ， <math display="block" id="M128"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><mn mathvariant="normal">2</mn><mi>n</mi><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419760&type= 48.00600052 8.29733276 <math display="block" id="M129"><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419762&type= 33.18933105 4.23333359 所以 <math display="block" id="M130"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></munderover><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mo stretchy="false">(</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419764&type= 31.07266808 8.29733276 <math display="block" id="M131"><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419766&type= 54.86400223 4.23333359 于是 <math display="block" id="M132"><mi>E</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">|</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow><mrow><mi mathvariant="bold-italic">a</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mi mathvariant="bold-italic">Q</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">t</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419768&type= 59.85932922 9.22866726 证毕。 2 威布尔-2阶几何分布 1 本节研究一类特殊的威布尔-离散位相型分布：威布尔-2阶几何分布。 d 阶几何分布是经典几何分布的推广，此问题最早被De Moivre [ 15 15 ] 提出。Fe11er [ 16 16 ] 把这一问题作为更新理论的一个应用。Philippou et al [ 17 17 ] 重新研究了这一问题，并给出 d 阶几何分布的定义。此后， d 阶几何分布便受到了广泛关注。下面回顾 d 阶几何分布的定义。 定义3 设在独立伯努利试验序列中，每次试验成功的概率为 <math id="M133"><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419771&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419770&type= 20.31999969 3.72533321 。记 <math id="M134"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419772&type= 1.86266661 3.47133350 为首次连续 d 次发生时所进行的试验次数，则称 <math id="M135"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419774&type= 1.86266661 3.47133350 服从的分布为 d 阶几何分布，简记为 <math id="M136"><msub><mrow><mi>G</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419776&type= 10.24466705 4.65666676 。 当 <math id="M137"><mi mathvariant="bold-italic">a</mi><mo>=</mo><msub><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1,0</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>×</mo><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419778&type= 29.80266571 4.48733330 ， <math id="M138"><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=3419781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419780&type= 15.15533447 3.64066648 ， <math display="block" id="M139"><mi mathvariant="bold-italic">Q</mi><mo>=</mo><msub><mrow><mfenced separators="|"><mrow><mtable><mtr><mtd><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mtd><mtd><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>p</mi></mtd></mtr></mtable></mtd><mtd><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtext> </mtext><mtable><mtr><mtd><mo>⋮</mo></mtd><mtd><mtext> </mtext><mo>⋮</mo></mtd></mtr></mtable></mtd><mtd><mo>⋮</mo></mtd><mtd><mtable><mtr><mtd><mo>⋱</mo></mtd><mtd><mo>⋮</mo></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mtd><mtd><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd><mtd><mtable columnalign="left"><mtr><mtd><mtable><mtr><mtd><mo>⋯</mo></mtd><mtd><mi>p</mi></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mo>⋯</mo></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></mrow><mrow><mi>d</mi><mo>×</mo><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419782&type= 54.77933502 28.02466774 时，离散位相型分布即为 d 阶几何分布。 当独立同分布 <math id="M140"><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∼</mo><mi mathvariant="normal">W</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=3419785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419784&type= 21.50533485 4.82599974 ， <math id="M141"><mi>ξ</mi><mo>~</mo><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419786&type= 14.98599911 4.65666676 时威布尔-离散位相型分布即为威布尔-2阶几何分布，简记为 <math id="M142"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419788&type= 20.74333382 4.82599974 。 由定理1知 <math id="M143"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419790&type= 20.74333382 4.82599974 分布的密度函数为 <math display="block" id="M144"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419792&type= 79.33266449 9.48266697 分布函数为 <math display="block" id="M145"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419794&type= 59.09733200 15.24000072 当 <math id="M146"><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419797&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419796&type= 8.72066689 3.72533321 时威布尔-2阶几何分布即为指数-2阶几何分布，当 <math id="M147"><mi>β</mi><mo>=</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419798&type= 8.72066689 3.72533321 时威布尔-2阶几何分布即为瑞利-2阶几何分布。下面将研究威布尔-2阶几何分布的密度函数、危险率函数的一些性质及其参数估计。 定理4 <math id="M148"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419800&type= 20.74333382 4.82599974 分布有以下性质： 1) 若 <math id="M149"><mn mathvariant="normal">0</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=3419803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419802&type= 15.32466698 4.06400013 ，则 <math id="M150"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419804&type= 20.74333382 4.82599974 分布的密度函数单调递减；若 <math id="M151"><mi>β</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419806&type= 8.72066689 3.81000018 ，则 <math id="M152"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419808&type= 20.74333382 4.82599974 分布的密度函数是单峰的。 2) <math id="M153"><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419810&type= 8.72066689 2.79399991 ，危险率函数 <math id="M154"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419812&type= 34.88266754 4.14866638 · <math id="M155"><mfrac><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow></mfrac><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419814&type= 54.01733780 9.48266697 当 <math id="M156"><mi>β</mi><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419816&type= 8.72066689 4.06400013 时， <math id="M157"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419818&type= 19.64266586 3.89466667 单减，当 <math id="M158"><mi>β</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419820&type= 8.72066689 3.81000018 时， <math id="M159"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419822&type= 19.64266586 3.89466667 的函数图像有不同的形状。且 <math display="block" id="M160"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>x</mi><mo>→</mo><msup><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></munder><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">∞</mi><mo>,</mo></mtd><mtd><mi>β</mi><mo><</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mi>α</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>,</mo></mtd><mtd><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>β</mi><mo>></mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419825&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419824&type= 62.31466675 21.25133324 <math display="block" id="M161"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>x</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mtable><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo></mtd><mtd><mi>β</mi><mo><</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn mathvariant="normal">2</mn><mi>α</mi><mo>,</mo></mtd><mtd><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mi mathvariant="normal">∞</mi><mo>,</mo></mtd><mtd><mi>β</mi><mo>></mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419827&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419826&type= 49.86866760 16.51000023 证 1) 对密度函数求导可得 <math display="block" id="M162"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup><mi>u</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>⋅</mo></mtd></mtr><mtr><mtd><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><mi>β</mi><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419828&type= 75.09933472 20.91266632 其中， <math id="M163"><mi>u</mi><mo>=</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419830&type= 11.76866722 2.87866688 ， <math id="M164"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mn mathvariant="normal">3</mn><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419832&type= 131.82600403 4.91066647 。 易知 <math id="M165"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419834&type= 54.94866943 4.06400013 ， <math id="M166"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419836&type= 8.80533314 4.48733330 <math id="M167"><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419838&type= 6.18066692 2.79399991 。若 <math id="M168"><mi>β</mi><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419840&type= 8.38199997 4.06400013 ，则 <math id="M169"><mi>f</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419842&type= 14.90133286 3.81000018 ，因此密度函数单调递减。若 <math id="M170"><mi>β</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419844&type= 8.72066689 3.81000018 ，令 <math display="block" id="M171"><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419846&type= 76.19999695 8.55133343 当 <math id="M173"><mi>u</mi><mo>→</mo><msup><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>+</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419848&type= 10.58333302 2.96333337 时， <math id="M174"><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>→</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419850&type= 13.46199894 3.55599999 ；当 <math id="M175"><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419853&type= 9.73666668 2.62466669 时， <math id="M176"><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419855&type= 15.24000072 3.55599999 ，故 <math id="M177"><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>β</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419857&type= 21.42066574 8.29733276 至少有一根。可证 <math id="M178"><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419859&type= 6.77333355 3.55599999 单调递增，于是 <math id="M179"><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419861&type= 6.77333355 3.55599999 有唯一根，记为 <math id="M180"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419863&type= 3.04799986 3.55599999 ，则 <math id="M181"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419865&type= 19.55799866 3.89466667 在 <math id="M182"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>/</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>β</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419867&type= 19.13466644 4.82599974 单调递增，在 <math id="M183"><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>/</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>β</mi></mrow></msup><mo>,</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419869&type= 21.59000015 4.82599974 单调递减。 图1 图1 给出了 <math id="M184"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419871&type= 20.74333382 4.82599974 分布参数取不同值时的密度函数图像。 2) 由(7)式和(8)式可得 <math id="M187"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419880&type= 19.64266586 3.89466667 ，对其求导 <math display="block" id="M188"><mtable columnalign="left"><mtr><mtd><mi>h</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>α</mi><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>⋅</mo></mtd></mtr><mtr><mtd><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo>-</mo><mi>β</mi><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419882&type= 75.09933472 20.91266632 其中, <math id="M189"><mi>u</mi><mo>=</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419884&type= 11.76866722 2.87866688 ， <math id="M190"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</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>u</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mi>p</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mn mathvariant="normal">2</mn><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419886&type= 98.72133636 4.91066647 。显然 <math id="M191"><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419889&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419888&type= 15.32466698 4.48733330 ，若 <math id="M192"><mi>β</mi><mo>≤</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419890&type= 8.72066689 4.06400013 ，则 <math id="M193"><mi>h</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419892&type= 15.32466698 3.55599999 ，因此危险率函数单调递减。 图2 图2 给出了 <math id="M194"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419894&type= 20.74333382 4.82599974 分布参数取不同值时的危险率函数图像。由 图2 图2 知，对不同的参数，危险率函数有不同的形状。证毕。 由定理2知，若 <math id="M197"><mi>X</mi><mo>∼</mo><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419903&type= 27.77066803 4.82599974 , <math id="M198"><mi>k</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419906&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419905&type= 8.38199997 2.79399991 ，则 <math id="M199"><mi>X</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419907&type= 2.62466669 2.62466669 的 <math id="M200"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419909&type= 1.69333339 2.79399991 阶原点矩为 <math id="M201"><mi>Ε</mi><mo stretchy="false">(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>α</mi></mrow><mrow><mo>-</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mi>Ψ</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo>,</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>,</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419911&type= 62.48400116 8.72066689 ,其中 <math id="M202"><mi>Ψ</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Γ</mi><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msup><mrow><mi>t</mi></mrow><mrow><mi>s</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>a</mi><mi>t</mi></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>z</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mi>z</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>t</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419913&type= 93.13333893 6.77333355 。 可以看出 <math id="M203"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419915&type= 20.74333382 4.82599974 分布的 <math id="M204"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419917&type= 1.69333339 2.79399991 阶矩的形式比较复杂，参数的矩估计不易求出。下面用极大似然估计法求参数的估计。 从 <math id="M205"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419919&type= 20.74333382 4.82599974 分布中抽取容量为 <math id="M206"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419921&type= 1.86266661 2.62466669 的简单随机样本 <math id="M207"><mfenced open="{" close="}" separators="|"><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419924&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419923&type= 28.87133217 4.57200003 ，记 <math id="M208"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419925&type= 21.67466736 4.48733330 <math id="M209"><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419927&type= 13.20799923 3.13266659 <math id="M210"><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419930&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419929&type= 3.47133350 3.55599999 为样本观测值，则其对数似然函数为 <math display="block" id="M211"><mi>l</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>;</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>α</mi><mi>β</mi><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419931&type= 74.42200470 8.29733276 <math display="block" id="M212"><mn mathvariant="normal">2</mn><mi>α</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo stretchy="false">)</mo></mrow><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419934&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419933&type= 58.25066376 8.29733276 <math display="block" id="M213"><mn mathvariant="normal">2</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419935&type= 64.09266663 8.29733276 令 <math display="block" id="M214"><mi>M</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">2</mn><mo>-</mo><mfrac><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow></mfrac><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419937&type= 51.39266586 9.56733322 <math display="block" id="M215"><mfrac><mrow><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419939&type= 51.73133087 9.56733322 ， <math display="block" id="M216"><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow></mfrac><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419941&type= 44.61933136 8.88999939 <math display="block" id="M217"><mfrac><mrow><mn mathvariant="normal">2</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419943&type= 51.73133087 9.56733322 则似然方程为： <math display="block" id="M218"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>l</mi></mrow><mrow><mo>∂</mo><mi>α</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>β</mi></mrow></msubsup></mrow><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419945&type= 74.42200470 8.29733276 <math display="block" id="M219"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>β</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>l</mi></mrow><mrow><mo>∂</mo><mi>β</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>β</mi></mrow></mfrac><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419947&type= 65.19332886 8.46666718 <math display="block" id="M220"><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup><mi>M</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419949&type= 23.79133415 4.65666676 <math display="block" id="M221"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>l</mi></mrow><mrow><mo>∂</mo><mi>p</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>n</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419951&type= 79.58666992 8.29733276 上述方程组的解即参数 <math id="M222"><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419953&type= 9.31333351 3.72533321 的极大似然估计。以下讨论 <math id="M223"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419955&type= 20.74333382 4.82599974 分布中两个参数已知时，另一个参数的极大似然估计的存在性、相合性以及渐近正态性。与文献[ 1 1 , 5 5 , 9 9 ]类似，可以得到如下定理。 定理 5 对 <math id="M224"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419957&type= 20.74333382 4.82599974 分布有： 1)如果 <math id="M225"><mi>β</mi><mo>,</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419959&type= 5.58799982 3.72533321 已知， <math id="M226"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419961&type= 1.94733334 2.62466669 未知，记 <math id="M227"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419964&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419963&type= 36.23733139 3.89466667 ， <math id="M228"><mi>Ι</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>α</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419965&type= 40.04733276 10.83733368 为Fisher信息量，则对 <math id="M229"><mo>∀</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419967&type= 4.14866638 2.96333337 ，方程(9)至少有一解 <math display="block" id="M230"><msub><mrow><mover accent="true"><mrow><mi>α</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419969&type= 6.60399961 3.97933340 <math display="block" id="M231"><mo stretchy="false">(</mo><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msup><mrow><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419971&type= 45.71999741 8.55133343 其中, <math id="M232"><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mi>i</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419973&type= 15.83266640 7.19666672 <math id="M233"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419975&type= 7.95866680 8.29733276 , <math id="M234"><msub><mrow><mover accent="true"><mrow><mi>α</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419977&type= 3.13266659 3.72533321 是参数 <math id="M235"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419979&type= 1.94733334 2.62466669 的相合估计和渐近正态估计，其渐近分布为 <math id="M236"><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mi>Ι</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419985&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419983&type= 20.06599998 8.12800026 。 2)若 <math id="M237"><mi>α</mi><mo>,</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419987&type= 5.58799982 3.47133350 已知， <math id="M238"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419990&type= 1.94733334 3.72533321 未知，记 <math id="M239"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419995&type= 36.32199860 3.89466667 ， <math id="M240"><mi>Ι</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>β</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3419998&type= 40.55533218 11.76866722 为Fisher信息量，则对 <math id="M241"><mo>∀</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420003&type= 4.14866638 2.96333337 ，方程(10)至少有一解 <math id="M242"><msub><mrow><mover accent="true"><mrow><mi>β</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><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=3420008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420006&type= 17.35666656 5.50333357 ， <math id="M243"><msub><mrow><mover accent="true"><mrow><mi>β</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420012&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420010&type= 3.13266659 5.50333357 是参数 <math id="M244"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420014&type= 1.94733334 3.72533321 的相合估计和渐近正态估计，其渐近分布为 <math id="M245"><mi>N</mi><mo stretchy="false">(</mo><mi>β</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mi>Ι</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420020&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420018&type= 21.08199883 8.46666718 。 3)若 <math id="M246"><mi>α</mi><mo>,</mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420022&type= 5.67266655 3.72533321 已知， <math id="M247"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420026&type= 1.86266661 3.47133350 未知，记 <math id="M248"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420030&type= 36.15266418 3.89466667 ， <math id="M249"><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420034&type= 41.82533264 11.68400002 为Fisher信息量，则对 <math id="M250"><mo>∀</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420038&type= 4.14866638 2.96333337 ，方程(11)在 <math id="M251"><msub><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>n</mi><mo><</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420043&type= 19.98133469 4.48733330 <math id="M252"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>+</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo stretchy="false">)</mo></mrow><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420047&type= 32.59666443 8.29733276 上至少有一解 <math id="M253"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420051&type= 3.04799986 4.91066647 (在 <math id="M254"><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420053&type= 3.64066648 3.81000018 上定义 <math id="M255"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420055&type= 13.54666710 4.91066647 )，则 <math id="M256"><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>P</mi><mo stretchy="false">(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420058&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420057&type= 22.43666649 4.57200003 ， <math id="M257"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420059&type= 3.04799986 4.91066647 是参数 <math id="M258"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420061&type= 1.86266661 3.47133350 的相合估计和渐近正态估计，其渐近分布为 <math id="M259"><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420063&type= 20.91266632 8.29733276 。 证 1) 若 <math id="M260"><mi>β</mi><mo>,</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420065&type= 5.58799982 3.72533321 已知，令 <math display="block" id="M261"><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mi>M</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420067&type= 41.31733322 8.29733276 ， <math display="block" id="M262"><mi>ω</mi><mi>'</mi><mo stretchy="false">(</mo><mi>α</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><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>β</mi></mrow></msubsup></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420069&type= 41.99466705 8.29733276 <math display="block" id="M263"><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow></mfrac><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420071&type= 31.83466721 8.80533314 <math display="block" id="M264"><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">8</mn><mi>p</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420073&type= 57.82733154 9.05933285 <math display="block" id="M265"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420075&type= 7.28133297 3.55599999 <math display="block" id="M266"><mi>p</mi><mo stretchy="false">)</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>β</mi></mrow></msubsup></mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420078&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420077&type= 22.01333427 8.29733276 <math display="block" id="M267"><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420079&type= 1.10066664 2.62466669 <math display="block" id="M268"><mfrac><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">6</mn><mi>p</mi><mo>-</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>u</mi></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>u</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420081&type= 58.92799759 8.55133343 易知 <math id="M269"><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420084&type= 7.78933334 3.55599999 对 <math id="M270"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420086&type= 1.94733334 2.62466669 单调递减且 <math id="M271"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>α</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420088&type= 20.40466690 4.57200003 ，则 <math display="block" id="M272"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420092&type= 24.12999916 4.82599974 <math display="block" id="M273"><mfrac><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>β</mi></mrow></msubsup></mrow><mo>-</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>α</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><msub><mrow/><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420094&type= 61.12933731 8.29733276 因此当 <math id="M274"><mi>α</mi><mo>></mo><mn mathvariant="normal">1</mn><mo>/</mo><mn mathvariant="normal">2</mn><msup><mrow><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420098&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420096&type= 17.44133377 3.47133350 时， <math id="M275"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420100&type= 29.04066658 4.82599974 。此外, <math display="block" id="M276"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>α</mi><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><mo>-</mo><mn mathvariant="normal">3</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">3</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>β</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420102&type= 58.92799759 8.55133343 , <math display="block" id="M277"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>β</mi></mrow></msup></mrow><mo>-</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>α</mi><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420104&type= 70.35800171 8.29733276 <math display="block" id="M278"><mfrac><mrow><mi>n</mi></mrow><mrow><mi>α</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>β</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420106&type= 44.53466797 8.55133343 当 <math id="M279"><mi>α</mi><mo><</mo><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msup><mrow><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420108&type= 39.87800217 8.55133343 时， <math id="M280"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420110&type= 29.04066658 4.82599974 。于是, <math id="M281"><mi>ω</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420112&type= 7.78933334 3.55599999 在区间 <math id="M282"><mo stretchy="false">(</mo><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><mi>p</mi><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msup><mrow><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mover><mrow><mi>x</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420114&type= 46.14333344 8.55133343 内至少存在一根。 容易验证 <math id="M283"><msub><mrow><mover accent="true"><mrow><mi>α</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420116&type= 3.13266659 3.72533321 满足文献[ 18 18 ]定理2.13的条件，故 <math id="M284"><msub><mrow><mover accent="true"><mrow><mi>α</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420118&type= 3.13266659 3.72533321 是 <math id="M285"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420120&type= 1.94733334 2.62466669 的相合估计。记参数 <math id="M286"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420122&type= 1.94733334 2.62466669 的真值为 <math id="M287"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420124&type= 3.13266659 3.55599999 ，为了验证 <math id="M288"><msub><mrow><mover accent="true"><mrow><mi>α</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420126&type= 3.13266659 3.72533321 的渐近正态性，可验证文献[ 18 18 ]定理2.14的4个条件： (a) 当 <math id="M289"><mi>x</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420128&type= 8.72066689 2.79399991 时， <math id="M290"><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>α</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420130&type= 16.67933464 7.53533316 ， <math id="M291"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420132&type= 18.03400040 7.70466709 ， <math id="M292"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420134&type= 18.03400040 7.70466709 都存在； (b) 对充分小的正数 <math id="M293"><mi>ϵ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420136&type= 1.69333339 2.79399991 ，当 <math id="M294"><mi>α</mi><mo>∈</mo><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>ϵ</mi><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420138&type= 29.71799850 4.48733330 时可知 <math display="block" id="M295"><mfenced open="|" close="|" separators="|"><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420141&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420140&type= 24.97666740 8.88999939 <math display="block" id="M296"><mfenced open="|" close="" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><msup><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>β</mi></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420142&type= 69.08799744 10.58333302 <math display="block" id="M297"><mn mathvariant="normal">2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>β</mi></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>⋅</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">7</mn><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420144&type= 62.90733337 4.48733330 <math display="block" id="M298"><mo stretchy="false">(</mo><mn mathvariant="normal">7</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420146&type= 69.68066406 4.48733330 <math display="block" id="M299"><mfenced open="" close="|" separators="|"><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfenced><mo>≤</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420148&type= 62.82266617 5.16466665 <math display="block" id="M300"><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>-</mo><mi mathvariant="normal">ϵ</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mn mathvariant="normal">6</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>β</mi></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">16</mn><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>β</mi></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></mrow></mfrac><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420150&type= 49.44533539 8.55133343 而 <math id="M301"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi>p</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420152&type= 44.87333298 6.77333355 (c) 令 <math id="M302"><mi>f</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>α</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420154&type= 31.32666779 7.53533316 ，对 <math id="M303"><mo>∀</mo><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420156&type= 10.83733368 2.96333337 ，总存在充分小的正数 <math id="M304"><msub><mrow><mi>ϵ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420158&type= 2.79399991 3.72533321 ，满足 <math id="M305"><mi>α</mi><mo>-</mo><msub><mrow><mi mathvariant="normal">ϵ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420160&type= 16.42533302 3.89466667 ，由中值定理知， <math id="M306"><mo>∃</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>∈</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mi>α</mi><mo>-</mo><msub><mrow><mi mathvariant="normal">ϵ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><mi>α</mi><mo>+</mo><msub><mrow><mi mathvariant="normal">ϵ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420162&type= 40.38600159 4.65666676 ，使得 <math display="block" id="M307"><mfenced open="|" close="|" separators="|"><mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi mathvariant="normal">ϵ</mi><mo stretchy="false">)</mo><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">ϵ</mi></mrow></mfrac></mrow></mfenced><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420164&type= 57.82733154 8.88999939 因为 <math display="block" id="M308"><mfenced open="|" close="|" separators="|"><mrow><mi>f</mi><mi>'</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mo stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420167&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420166&type= 75.86133575 4.82599974 <math display="block" id="M309"><mn mathvariant="normal">4</mn><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>+</mo><mn mathvariant="normal">3</mn><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420168&type= 38.69266510 4.91066647 <math display="block" id="M310"><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420171&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420170&type= 32.59666443 4.31799984 <math display="block" id="M311"><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420172&type= 33.95133591 4.57200003 <math display="block" id="M312"><mo stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420174&type= 39.87800217 5.58799982 <math display="block" id="M313"><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mn mathvariant="normal">2</mn><mi>p</mi><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420176&type= 23.53733444 4.31799984 <math display="block" id="M314"><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420178&type= 32.59666443 4.31799984 <math display="block" id="M315"><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><msubsup><mrow><mi>α</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420180&type= 29.63333321 4.57200003 <math display="block" id="M316"><mo><</mo><mfrac><mrow><mn mathvariant="normal">5</mn><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420182&type= 20.82799911 8.55133343 <math display="block" id="M317"><mfrac><mrow><mn mathvariant="normal">3</mn><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mi>β</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>β</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow></mfrac><mo>=</mo><mo>:</mo><msub><mrow><mi>K</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=3420185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420184&type= 43.94200134 8.55133343 易证 <math id="M318"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420186&type= 27.85533333 6.77333355 ，所以由控制收敛定理，对 <math id="M319"><mo>∀</mo><mi>α</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420188&type= 10.83733368 2.96333337 <math display="block" id="M320"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mfrac><mrow><mo>∂</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>α</mi></mrow></mfrac></mrow><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><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>ϵ</mi><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder></mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi>ϵ</mi></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>ϵ</mi><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>-</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi>ϵ</mi></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>ϵ</mi><mo>→</mo><mn mathvariant="normal">0</mn></mrow></munder><mfrac><mrow><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mi>f</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo>+</mo><mi>ϵ</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>f</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>x</mi></mrow><mrow><mi>ϵ</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420190&type= 73.49066925 38.77733231 同理有 <math id="M321"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420192&type= 31.58066750 7.70466709 。 (d) 对充分小的正数 <math id="M322"><mi>ϵ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420194&type= 1.69333339 2.79399991 ，当 <math id="M323"><mi>α</mi><mo>∈</mo><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>ϵ</mi><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420196&type= 29.71799850 4.48733330 时，Fisher信息量为 <math display="block" id="M324"><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>α</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420198&type= 33.10466385 10.83733368 <math display="block" id="M325"><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>α</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup><mo>-</mo><mi>p</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo><</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420200&type= 119.37999725 12.61533356 <math display="block" id="M326"><mi>E</mi><mfenced separators="|"><mrow><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">ϵ</mi></mrow></mfrac><mo>+</mo><mn mathvariant="normal">2</mn><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">6</mn><msup><mrow><mi>X</mi></mrow><mrow><mi>β</mi></mrow></msup></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo><</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420202&type= 61.38333511 11.68400002 又因为 <math id="M327"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420204&type= 20.74333382 4.82599974 分布为非退化分布，故 <math id="M328"><mi>I</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420206&type= 13.54666710 3.55599999 。于是 <math id="M329"><msub><mrow><mover accent="true"><mrow><mi>α</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420208&type= 3.13266659 3.72533321 是 <math id="M330"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420210&type= 1.94733334 2.62466669 的渐近正态估计，且其渐近分布为 <math id="M331"><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420212&type= 20.48933411 8.29733276 。 2)和3)证明过程与1)类似。 证毕。 3 威布尔-2阶几何分布的数值模拟和实例 1 3.1 数值模拟 2 本节考虑两参数已知情况下，另一个参数的极大似然及其区间估计。由于方法类似，下面只考虑 <math id="M332"><mi>α</mi><mo>,</mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420214&type= 5.67266655 3.72533321 已知， <math id="M333"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420216&type= 1.86266661 3.47133350 未知时的情况。 设 <math id="M334"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420218&type= 38.77733231 4.48733330 为来自 <math id="M335"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420220&type= 17.35666656 4.82599974 <math id="M336"><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420222&type= 3.47133350 3.72533321 分布的观测数据，由定理5知，参数 <math id="M337"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420224&type= 1.86266661 3.47133350 的极大似然估计 <math id="M338"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420226&type= 3.04799986 4.91066647 近似服从正态分布 <math id="M339"><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420228&type= 20.91266632 8.29733276 ，因Fisher信息量 <math id="M340"><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mo>-</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420230&type= 42.33333206 8.55133343 的计算较为复杂，由大数定律知 <math display="block" id="M341"><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mover accent="false"><mo stretchy="true">→</mo><mrow><mi>P</mi></mrow></mover><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420232&type= 63.24599838 8.55133343 又因 <math id="M342"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420234&type= 9.99066734 4.91066647 ，故可用 <math id="M343"><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420236&type= 9.65199947 4.91066647 代替 <math id="M344"><mi>Ι</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420239&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420238&type= 7.62000036 3.72533321 ，于是 <math id="M345"><mo stretchy="false">[</mo><mi>n</mi><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420240&type= 18.37266731 5.16466665 给出了参数 <math id="M346"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420242&type= 1.86266661 3.47133350 的极大似然估计的渐近方差(可记为 <math id="M347"><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420244&type= 12.27666664 4.31799984 )，则有 <math display="block" id="M348"><mo stretchy="false">(</mo><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>=</mo><mi>n</mi><msub><mrow><mi>J</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mfenced open="" close="|" separators="|"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mi>p</mi><mo>=</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></mrow></msub><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420246&type= 76.45400238 11.00666714 <math display="block" id="M349"><mfrac><mrow><mn mathvariant="normal">2</mn><mi>n</mi></mrow><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msubsup><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow><mrow><msup><mrow/><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msubsup><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><mo>-</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>α</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>β</mi></mrow></msubsup></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420248&type= 119.63400269 10.66799927 未知参数 <math id="M350"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420250&type= 1.86266661 3.47133350 的95%置信区间为 <math id="M351"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>±</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0.975</mn></mrow></msub><mroot><mrow><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420252&type= 31.32666779 5.67266655 。 记 <math id="M352"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420254&type= 1.86266661 2.62466669 为样本容量， <math id="M353"><mi>θ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420256&type= 18.88066673 3.89466667 ，则可通过以下步骤得到参数 <math id="M354"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420258&type= 1.86266661 3.47133350 的区间估计： 步骤1 产生 <math id="M355"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420260&type= 1.86266661 2.62466669 个服从 <math id="M356"><mi>U</mi><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=3420263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420262&type= 12.19199944 3.55599999 分布的随机数，记为 <math id="M357"><mi>u</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420264&type= 23.02933502 4.48733330 ； 步骤2 通过计算知 <math id="M358"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420267&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420266&type= 13.20799923 4.48733330 分布的分位数函数为 <math display="block" id="M359"><msup><mrow><mi>F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><mi>u</mi><mo>;</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420268&type= 19.89666748 3.81000018 <math display="block" id="M360"><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>p</mi><mo>+</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">4</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>u</mi></mrow></mfrac><mo>-</mo><mn mathvariant="normal">3</mn><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><mi>p</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow><mrow><mi>α</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420270&type= 61.04467010 28.44799995 将 <math id="M361"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420272&type= 13.80066681 3.97933340 分别代入上式，得 <math id="M362"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>;</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420274&type= 43.94200134 4.65666676 ，则 <math id="M363"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">s</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><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><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420276&type= 26.83933258 4.48733330 为来自 <math id="M364"><mi mathvariant="normal">W</mi><msub><mrow><mi mathvariant="normal">G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420278&type= 13.20799923 4.48733330 分布的容量为 <math id="M365"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420280&type= 1.86266661 2.62466669 的样本； 步骤3 给定参数的初始值 <math id="M366"><msup><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420282&type= 2.96333337 3.72533321 ，得到参数 <math id="M367"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420284&type= 1.86266661 3.47133350 的极大似然估计 <math id="M368"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420286&type= 3.04799986 4.91066647 ，并判断真值 <math id="M369"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420288&type= 1.86266661 3.47133350 是否落入95%置信区间 <math id="M370"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo>±</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0.975</mn></mrow></msub><mroot><mrow><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420290&type= 31.32666779 5.67266655 ； 步骤4 重复步骤1~3 <math id="M371"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420292&type= 2.87866688 2.70933342 次, 将得到的 <math id="M372"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420294&type= 2.87866688 2.70933342 个估计值的均值作为参数 <math id="M373"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420296&type= 1.86266661 3.47133350 的极大似然估计， <math id="M374"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420298&type= 2.87866688 2.70933342 次模拟得到的置信下(上)界的均值作为参数 <math id="M375"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420300&type= 1.86266661 3.47133350 的置信下(上)界。 假设模拟得到的极大似然估计为 <math id="M376"><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420302&type= 27.43200111 4.91066647 ，相应的95%置信区间简记为 <math id="M377"><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>±</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0.975</mn></mrow></msub><mroot><mrow><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420304&type= 44.78866577 7.19666672 ，若真值 <math id="M378"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420306&type= 1.86266661 3.47133350 落入该区间记 <math id="M379"><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420308&type= 16.67933464 4.06400013 ，否则 <math id="M380"><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo stretchy="false">(</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=3420311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420310&type= 16.67933464 4.06400013 ，记 <math id="M381"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420312&type= 2.87866688 2.70933342 个估计值的均值为 <math display="block" id="M382"><mi mathvariant="normal">M</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420315&type= 35.30599976 8.29733276 标准差为 <math display="block" id="M383"><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">M</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">N</mi><msup><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420317&type= 58.75866699 9.65199947 标准误差为 <math display="block" id="M384"><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mroot><mrow><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420319&type= 41.40200043 8.29733276 参数 <math id="M385"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420321&type= 1.86266661 3.47133350 的95%平均置信下界为 <math display="block" id="M386"><mi mathvariant="normal">L</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">L</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0.975</mn></mrow></msub><mroot><mrow><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420323&type= 62.65333557 8.29733276 参数 <math id="M387"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420325&type= 1.86266661 3.47133350 的95%平均置信上界为 <math display="block" id="M388"><mi mathvariant="normal">U</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">L</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0.975</mn></mrow></msub><mroot><mrow><mover accent="true"><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="true">̂</mo></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo stretchy="true">̂</mo></mover></mrow><mrow><mi>n</mi><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420327&type= 63.16133499 8.29733276 参数 <math id="M389"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420329&type= 1.86266661 3.47133350 的95%置信区间的覆盖率为 <math display="block" id="M390"><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>N</mi></mrow></munderover><mrow><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420331&type= 35.64466476 8.29733276 本文 <math id="M391"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420333&type= 2.87866688 2.70933342 取1 000， 表1 表1 给出了样本量分别100，1 000，10 000时的模拟结果。 <math id="M402"><mi>n</mi></math> <math id="M403"><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></math> <math id="M404"><mi>p</mi></math> <math id="M405"><msup><mrow><mi>p</mi></mrow><mrow><mn>0</mn></mrow></msup></math> <math id="M406"><mi mathvariant="normal">M</mi><mi mathvariant="normal">E</mi><mi mathvariant="normal">A</mi><mi mathvariant="normal">N</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> <math id="M407"><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> <math id="M408"><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> <math id="M409"><mi mathvariant="normal">L</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">L</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> <math id="M410"><mi mathvariant="normal">U</mi><mi mathvariant="normal">C</mi><mi mathvariant="normal">L</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> <math id="M411"><mi mathvariant="normal">C</mi><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> (1,1) 0.25 0.50 0.252 0 0.023 7 0.024 1 0.204 8 0.299 3 0.961 0 100 (1,2) 0.50 0.75 0.501 3 0.050 8 0.050 9 0.401 5 0.601 0 0.950 0 (2,2) 0.75 0.50 0.753 5 0.080 3 0.079 2 0.594 8 0.908 7 0.952 0 (1,1) 0.25 0.50 0.249 9 0.007 5 0.007 6 0.235 0 0.264 7 0.954 0 1 000 (1,2) 0.5 0.75 0.499 8 0.015 9 0.016 0 0.468 4 0.531 2 0.948 0 (2,2) 0.75 0.50 0.749 9 0.024 5 0.025 0 0.701 0 0.798 9 0.951 0 (1,1) 0.25 0.50 0.250 0 0.002 4 0.002 4 0.245 3 0.254 7 0.963 0 10 000 (1,2) 0.50 0.75 0.500 0 0.005 2 0.005 1 0.490 1 0.509 9 0.950 2 (2,2) 0.75 0.50 0.750 0 0.007 9 0.007 9 0.734 5 0.765 4 0.951 0 从 表1 表1 可以看出：参数 <math id="M392"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420345&type= 1.86266661 3.47133350 的标准差 <math id="M393"><mi mathvariant="normal">S</mi><mi mathvariant="normal">D</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420347&type= 10.83733368 3.72533321 、标准误差 <math id="M394"><mi mathvariant="normal">S</mi><mi mathvariant="normal">E</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420349&type= 10.58333302 3.72533321 都接近于0，且两者相差很小，表明参数 <math id="M395"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420351&type= 1.86266661 3.47133350 的极大似然估计具有优良的有限样本性质；参数 <math id="M396"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420353&type= 1.86266661 3.47133350 的95%置信区间的覆盖率稳定在0.95附近，且随着样本量的增加，参数 <math id="M397"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420355&type= 1.86266661 3.47133350 的95%置信上界和置信下界越来越接近参数真值，表明该区间估计方法有效。 为了考察区间估计的稳定性与合理性，下面我们引入模拟次数-覆盖率关系图，该图横轴表示模拟的次数，纵轴表示参数的95%置信区间的覆盖率。 图3 图3 给出了样本量为1 000，模拟次数为10 000，参数 <math id="M412"><mi>α</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">2</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420357&type= 33.27399826 3.72533321 时参数 <math id="M413"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420359&type= 1.86266661 3.47133350 的95%置信区间的覆盖率图像。从 图3 图3 可以看出模拟次数对参数 <math id="M414"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420361&type= 1.86266661 3.47133350 的95%置信区间覆盖率影响不大，当模拟次数大于3 000时覆盖率趋于稳定，且覆盖率维持在95%左右，与该置信区间的置信水平95%非常接近。 3.2 实例研究 2 本节使用威布尔-2阶几何分布拟合两组真实数据，以此来说明其应用。 例1：13架波音720喷气式飞机的空调系统连续两次故障时间间隔(小时)的数据集 [ 19 19 ] 。 例2：Minneapolis/St Paul 地区连续30年3月份降水量(英寸)的数据集 [ 20 20 ] 。 表2 表2 给出了分别用威布尔-2阶几何分布，指数-2阶几何分布，威布尔分布去拟合真实数据得到的极大似然估计值，K-S 统计量值，以及相应的 <math id="M417"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420370&type= 1.86266661 3.47133350 值，从 表2 表2 中可以看出：对第一组数据，三种分布都能较好的拟合；对第二组数据，除指数-2阶几何分布外都能较好的拟合，但相比而言威布尔-2阶几何分布有更小的K-S统计量和更大的 <math id="M418"><mi>p</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420372&type= 1.86266661 3.47133350 值，具有更好的拟合效果。 例子 分布 参数估计值 K-S统计量 <math id="M425"><mi>p</mi></math> 值 WG 2 (0.000 7,1.208 8,0.373 6) 0.039 2 0.899 8 例1( n =213) EG 2 (0.003 8,0.679 4) 0.048 4 0.699 9 Weibull (0.015 7,0.924 6) 0.052 0 0.613 0 WG 2 (0.032 0,2.401 5,0.348 2) 0.063 5 0.999 1 例2( n =30) EG 2 (0.298 6,0.999 9) 0.236 6 0.058 3 Weibull (0.315 5,1.808 9) 0.067 5 0.997 7 图4 图4 , 图5 图5 给出了两组数据的分布直方图以及分别用三种分布拟合的效果图，其结果与 表2 表2 相似。 当威布尔-2阶几何分布被选做拟合模型时,我们感兴趣的是它能否退化为指数-2阶几何分布。 对例1和例2进行似然比检验,结果如下: 例1 <math id="M419"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>:</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420382&type= 14.73199940 4.65666676 <math id="M420"><mi>v</mi><mi>s</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420384&type= 5.07999992 2.62466669 <math id="M421"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>β</mi><mo>≠</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn mathvariant="normal">0.0752</mn><mtext> </mtext><mn mathvariant="normal">7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420386&type= 36.91466904 4.82599974 例2 <math id="M422"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>:</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420389&type= 14.13933372 4.65666676 <math id="M423"><mi>v</mi><mi>s</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420391&type= 4.14866638 2.62466669 <math id="M424"><mtext> </mtext><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>β</mi><mo>≠</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>p</mi><mo>=</mo><mn mathvariant="normal">1.294</mn><mtext> </mtext><mn mathvariant="normal">1</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420393&type= 44.45000076 5.07999992 对例1而言,在5%显著性水平下不能够拒绝原假设,即对第一组数据威布尔-2阶几何分布能够退化为指数-2阶几何分布。对例2而言,在5%显著性水平下能够拒绝原假设,即对第二组数据威布尔-2阶几何分布不能够退化为指数-2阶几何分布。 4 结 语 1 基于随机最小值分布产生机制，提出了一类新的分布:威布尔-离散位相型分布，研究了其统计性质，并重点关注了威布尔-2阶几何分布。推导了威布尔-2阶几何分布参数的极大似然估计的相合性与渐近正态性，从理论上证明了新分布的优良性质。除此之外，本文也引入了置信区间覆盖率指标、模拟次数-覆盖率图对区间估计的效果进行了考察,证实了威布尔-2阶几何分布具有较好的有限样本性质。在实例研究中，用威布尔-2阶几何分布拟合真实数据，发现其具有较好的拟合效果。 论文中仍有一些问题值得进一步研究。如在研究参数极大似然估计的渐近正态性时，由于对数似然函数2阶偏导的形式复杂且其条件期望不易求出，本文只给出了两个参数已知时另一个参数的渐近正态性，并未证明三参数的渐近正态性。另外，由于离散位相型分布是一类包含范围特别广的分布，通过选取不同的 <math id="M426"><mi mathvariant="bold-italic">a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420395&type= 1.94733334 2.62466669 和 <math id="M427"><mi mathvariant="bold-italic">Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=3420397&type= 2.96333337 3.04799986 就能得到不同的分布，然后将其与不同的连续分布进行混合，就能得到更多的混合分布，使其能更好地拟合真实数据，有兴趣的学者可进一步探讨。 参考文献 ADAMIDIS K , LOUKAS S . 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DOI: 10.2307/2346869 http://dx.doi.org/10.2307/2346869 . On quick choice of power transformation 10.14188/j.1671-8836.2018.0108 O211.3 A 1671-8836（2020）03-0215-09 ZHANG Zhifang,LIU Luqin.Weibull-Discrete Phase-Type Distribution [J]. J Wuhan Univ （ Nat Sci Ed ）,2020,66(3):215-223. DOI: 10.14188/j.1671-8836.2018.0108 http://dx.doi.org/10.14188/j.1671-8836.2018.0108 (Ch). ZHANG Zhifang,LIU Luqin.Weibull-Discrete Phase-Type Distribution [J].J Wuhan Univ（Nat Sci Ed）,2020,66(3):215-223. DOI:10.14188/j.1671-8836.2018.0108(Ch). 张志方,刘禄勤.威布尔⁃离散位相型分布[J].武汉大学学报（理学版）,2020,66(3):215-223. DOI:10.14188/j.1671-8836.2018.0108 张志方,刘禄勤.威布尔⁃离散位相型分布[J].武汉大学学报（理学版）,2020,66(3):215-223. DOI:10.14188/j.1671-8836.2018.0108 国家自然科学基金（1171263） 2018-04-15 2020-06-24 2020-09-30 武汉大学学报（理学版） 3 66
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