一类广义指数分布的统计推断及其推广研究

【作者】 张娟；

【导师】 徐东胜；

【作者基本信息】 西南石油大学 ， 数学， 2019， 硕士

【摘要】 可靠性数理统计问题在分析产品寿命、可靠性及失效律等方面具有十分重要的理论指导与实践意义。指数分布是可靠性数理统计中的一个重要分布,然而指数分布无记忆性这一特点使它在实际的应用中有所约束,广义指数分布作为指数分布的一个延伸,克服了这一特点,从而使其广泛的应用到可靠性问题分析中。在前人的工作基础上,本文提出了一个新的两参数广义指数分布GE(x;α,β),并对该分布进行了统计推断和推广。具体工作如下:1、提出一类新的广义指数分布并讨论其统计特征。从单调性、有界性和右连续性三个方面入手,证明了新构造的函数是一个分布,明确该分布的数理定义;讨论了不同情况下该分布的概率密度函数和失效率函数的单调性,并刻画出相应的函数图像;研究了该分布的简单数字特征,如k阶矩的存在性证明、一阶矩和二阶矩的具体表达式、次序统计量及其相关性定理证明。2、全样本场合下,讨论了该分布的五种参数估计。矩估计和极大似然估计分别讨论了三种情况:位置参数已知而形状参数未知、位置参数未知而形状参数已知、位置参数和形状参数均已知;另外用最佳线性无偏估计、最优同变估计以及区间估计对该分布进行了估计。3、对GE分布的系数从特殊性向一般性进行推广,推广的GE1分布更具普适性。同样地,采用单调性、有界性、右连续性三个定理验证它确实是一个分布,并得出两个推广系数之间的关系k2=k1+1;随后证明了推广分布k阶矩的存在性,给出了一阶矩、二阶矩和方差表达式,并用这些表达式证明了GE分布和GE1分布之间互化关系的正确性。最后,对推广分布做参数估计,得到了矩估计和极大似然估计的参数表达形式,以及在一些特定情况下的参数估计表达式。4、对推广的GE1分布进行再推广,得到原分布的一般式——GE2分布。首先给出该分布的定义,刻画了概率密度函数的图像;其次从矩的方面研究了其数字特征,说明了原分布是GE2分布n=1时的一种特殊分布;最后用矩估计和极大似然估计对该分布进行了参数估计。文末对该两参数广义指数分布的统计性质及其推广进行了归纳概括,给出了本文的结论。并提出了几个未来有可能完成的、更深入的研究方向。更多还原

【Abstract】 Reliability mathematical statistics has very important theoretical guidance and practical significance in the analysis of product life,reliability and failure law.Exponential distribution is an important distribution in reliability mathematical statistics.Howeveir,the memorability of exponential distribution restricts its application in practice.Generalized exponential distribution,as an extension of exponential distribution,overcomes the characteristic of memory-free,so that it can be widely used in reliability analysis.This paper studies a kind of generalized exponential distribution.Reliability mathematical statistics is of great theoretical and practical significance in the analysis of product life,reliability and failure law.Exponential distribution is an important distribution in reliability mathematical statistics,but its characteristic of memoryless makes it restricted in practical application.Generalized exponential distribution,as an extension of exponential distribution,overcomes this characteristic and makes it widely used in reliability analysis.On the basis of previous work,a new two-parameter generalized exponential distribution GE in this paper is proposed,and then inferred and generalized by statistics.The specific work is as follows:A new kind of generalized exponential distribution and its statistical characteristics are proposed and discussed.Starting from three theorems of monotonicity,boundedness and right continuity,it is proved that the new function is a distribution and its mathematical definition is clearly defined.The monotonicity of probability density function and failure rate function of the distribution under different conditions is discussed,and the corresponding function image is depicted.Simple numerical characteristics of the distribution are studied,such as the existence proof of k moments,the concrete expressions of first and second moments,order statistics and proof of correlation theorems.In the case of full sample,five parameter estimates of the distribution are discussed.Moment estimation and maximum likelihood estimation are discussed respectively in three cases:known position parameter with unknown shape parameter,unknown position parameter with known shape parameter,known position parameter with known shape parameter.In addition,the distribution is estimated by the optimal unbiased estimation,the optimal covariant estimation and the interval estimation.GE is generalized to a more universal distribution GE1.Similarly,three theorems of monotonicity,boundedness and right continuity are used to verify that GE1 is indeed a distribution and the relationship between k 1and k2 is k2=k1+1.Then k moment of GE1 is proved to be existent,and the expressions of first moment,second moment and variance are given.These expressions are used to prove the correctness of the interdependence between GE and GE1.Finally,the parameter estimation of GE1 is estimated to obtain the expressions of moment estimation,maximum likelihood estimation,and the expression of parameter estimation in some specific cases.After generalizing GE1,GE2 is obtained.Firstly,the definition of GE2,the image of probability density function are given.Secondly,the digital characteristics are studied from the aspect of moment and explain that GE is a special distribution of GE2 while n=1.Finally,the parameter estimation of distribution is estimated by moment estimation and maximum likelihood estimation.At the end of this paper,the statistical properties of GE1 and GE2 are summarized and the conclusions of this paper are given.Several possible further research directions are proposed.更多还原