两类单参数分布的二次贝叶斯估计

发布时间：2018-03-15 23:30

本文选题：Pareto分布　切入点：均匀分布　出处：《北京交通大学》2017年硕士论文　论文类型：学位论文

【摘要】：Pareto分布和均匀分布由于其应用广泛一直受到研究者的重视,提出了多种参数估计方法,其中常用的有:极大似然估计、无偏估计和贝叶斯估计等等。大样本情况下,这几种方法都能够获得准确的结果。而小样本情况下,我们通常使用贝叶斯估计方法,但贝叶斯估计方法在计算参数估计时,常常存在积分困难的情况,若使用Gibbs抽样方法,运算时计算量巨大,给参数估计带来麻烦。在此基础上,本文提出了一种新的参数估计——二次贝叶斯估计。文章中,在定义统计量T的基础上,引入统计量T2,构造出参数的二次贝叶斯估计。二次贝叶斯估计既采纳了先验信息,同时避免了计算繁琐的后验期望。文章首先通过计算获得这两类分布的二次贝叶斯估计的显式解;其次在均方误差准则下,证明了二次贝叶斯估计优于极大似然估计与一致最小方差无偏估计。数值模拟比较发现,无论先验的选择和先验分布的参数取值如何变化,二次贝叶斯估计与贝叶斯估计均非常相近。随着样本容量n的增大,二次贝叶斯估计向贝叶斯估计趋近;同时,随着先验信息的集中,二次贝叶斯估计也向贝叶斯估计趋近。综上所述,二次贝叶斯估计对Pareto分布和均匀分布都是有效的。
[Abstract]:Because of its wide application, Pareto distribution and uniform distribution have been paid much attention to by researchers. A variety of parameter estimation methods are proposed, such as maximum likelihood estimation, unbiased estimation, Bayesian estimation and so on. In the case of small sample, we usually use Bayesian estimation method, but Bayesian estimation method often has difficulty in calculating parameter estimation, if we use Gibbs sampling method, In this paper, a new parameter estimator, quadratic Bayesian estimation, is proposed. In this paper, based on the definition of statistical quantity T, a new parameter estimation is proposed. The quadratic Bayesian estimation of parameters is constructed by introducing the statistic T2. The quadratic Bayesian estimation not only adopts the prior information, At the same time, the complicated posteriori expectation is avoided. Firstly, the explicit solution of the quadratic Bayesian estimation of these two kinds of distributions is obtained by calculation; secondly, under the mean square error criterion, It is proved that the quadratic Bayesian estimation is superior to the maximum likelihood estimation and the uniform minimum variance unbiased estimation. The quadratic Bayesian estimation is very similar to the Bayesian estimation. With the increase of the sample size n, the quadratic Bayesian estimation approaches to the Bayesian estimation, at the same time, with the concentration of prior information, Quadratic Bayesian estimation also approaches to Bayesian estimation. In conclusion, quadratic Bayesian estimation is effective for both Pareto distribution and uniform distribution.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212.8

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