Entropy损失函数下若干分布参数的Bayes估计及其性质

发布时间：2017-12-27 04:31

本文关键词：Entropy损失函数下若干分布参数的Bayes估计及其性质　出处：《兰州交通大学》2015年硕士论文　论文类型：学位论文

【摘要】：随着科学技术的日新月异和社会的向前发展,Bayesian统计方法作为一种新型的统计方法逐渐发展起来,它和经典统计方法一起构成了概率统计的重要部分.相对于经典统计方法而言,Bayesian统计方法能够尽可能多的综合各类信息进行统计推断,并且能够更加有效地结合实际情况,即考虑后果和效益问题,在实际生活的各个方面都发挥着越来越重要的作用.要考虑后果,就得引入损失函数,损失函数作为衡量统计决策的重要尺度,在统计推断中发挥着非常重要的作用.要进行统计推断,最重要的问题就是进行参数估计,而参数估计的相关性质可从统计决策的角度来分析估计的优良性,从而有助于我们做出正确合理的统计推断.由于指数-威布尔(EW)分布、对数正态(LN)分布以及帕累托(Pareto)分布三大常用分布在工程学、可靠性以及经济学方面的广泛应用,本文主要在非对称Entropy损失函数下研究了这三大分布参数的Bayes估计及其相关性质,并给出了相应的实际应用及实例模拟.首先,主要介绍了本文的研究背景、目的及其意义,并对Bayes理论的发展过程进行了简要介绍,指出了经典统计方法的局限性以及Bayes统计方法利用已有的先验信息、样本信息和总体信息进行参数估计的优良性质;接着介绍了Bayes方法以及经验Bayes(EB)方法的基本理论,并指出了常用对称损失函数的不足;然后介绍了非对称Entropy损失函数的研究现状以及根据LINEX损失函数和Entropy损失函数之间的关系将其转化为LINEX损失函数的方法和利用已有结论研究问题的思想.其次,在Entropy损失函数下,当先验分布已知时,讨论了指数-威布尔分布、对数正态分布、帕累托分布参数在共轭先验、Jeffrey’s先验、多层先验下的Bayes估计,并利用Monte Carlo方法进行了随机模拟;当先验分布未知时,利用核密度估计方法构造了这三大分布参数的经验Bayes估计,并利用了()2()1()a b a be a b e e----≤-(其中a0,b0)不等式、Holder不等式和Markov不等式证明了该EB估计的渐进最优性,利用反证法证明了该EB估计的可容许性;当两参数均未知时,根据Lindley逼近定理给出了帕累托分布参数的Bayes估计的具体表达形式.最后,在Entropy损失函数下,将指数-威布尔分布的可靠度R和失效率λ看成是随机变量或者是参数θ的函数,利用密度函数的相关性质求出了可靠度R和失效率λ后验密度,然后讨论了指数-威布尔分布的可靠度R和失效率λ的Bayes估计及其极大似然估计,并利用Monte Carlo方法进行了随机模拟,结果表明:Entropy损失函数下的Bayes估计较极大似然估计和平方损失函数下的Bayes估计精度高.然后应用Bayes估计讨论了对数正态分布场合下产品加固性能的Bayes评估,并利用实例证明了本文给出的评估方法是合理的,且文中给出的未知参数先验分布的取法是可行的.
[Abstract]:Along with the development of science and technology and social change rapidly, the Bayesian statistical method as a new statistical method developed gradually, and classical statistical methods together constitute an important part of probability and statistics. Compared with the classical statistical method, Bayesian statistical method can synthesize the information for statistical inference as much as possible, and more the effective combination of the actual situation, consider the consequences and benefits, are playing an increasingly important role in various practical aspects of life. To consider the consequences, we must introduce the loss function, the loss function as an important measure of statistical decision, plays a very important role in the statistical inference. To perform statistical inference the most important problem is to estimate the parameters, and the related properties of the parameter estimation from statistical decision analysis and estimation from benign. Which helps us to make correct statistical inference. (EW) due to exponential Weibull distribution and lognormal distribution (LN) and Pareto (Pareto) is widely used in the distribution of three commonly used distribution engineering, reliability and economics, this paper mainly Bayes estimation in asymmetric Entropy loss function on the three the distribution parameters and its properties, and gives the simulation of the actual application and corresponding examples. First, mainly introduced the research background, purpose and significance of this paper, and the development process of Bayes theory are briefly introduced, and pointed out the limitations of the classical statistical method and Bayes statistical method using prior information, the sample information and the overall information of the excellent properties of the parameter estimation; then introduces the methods and experience of Bayes Bayes (EB) theory, and points out the common symmetric loss function The number of problems; and then introduces the research status of asymmetric Entropy loss function, according to the relationship between LINEX loss function and Entropy loss function into LINEX loss function and using the existing research conclusion problem thought. Secondly, under Entropy loss function, when the prior distribution is known, discussed the estimation of Bayes index Weibull distribution and lognormal distribution, Pareto distribution parameters in the conjugate prior and Jeffrey 's prior, multilayer prior, and has carried on the simulation using Monte Carlo method; when the distribution is unknown a priori estimation, empirical Bayes method constructs the three distribution parameters using the kernel density, and the use of () 2 (1) (a) B a be a B e e---- - (A0, B0) inequality and Holder inequality and Markov inequality proved that the EB estimation of asymptotic optimality, using inverse method proves that the EB. The admissibility of two; when the parameters are unknown, according to the Lindley approximation theorem gives the estimation of Bayes parameters of the Pareto distribution of specific forms of expression. Finally, under Entropy loss function, reliability and failure rate of the lambda R exponential Weibull distribution as a random variable or function parameters, using the related the nature of density function obtained by the R reliability and failure rate of a posterior density, and then discusses the reliability index of R - Weibull distribution and failure rate of lambda Bayes estimation and maximum likelihood estimation, and stochastic simulation, using the Monte Carlo method the results show that the Bayes Entropy loss function estimation is Bayes the maximum likelihood estimation and square loss function under high estimation accuracy. Then the application of Bayes estimation is discussed Bayes evaluation under lognormal distribution are strengthening performance, and use examples to prove this to the The method of evaluation is reasonable, and it is feasible to take the prior distribution of unknown parameters given in this paper.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O212.8

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