正态分布参数的线性结构的贝叶斯估计

发布时间：2018-06-26 11:46

本文选题：正态分布 + 线性结构贝叶斯估计　；参考：《北京交通大学》2017年硕士论文

【摘要】：正态分布在数理统计学的理论研究和实际应用中具有十分重要的地位,越来越多的研究领域都涉及到正态分布的参数估计问题,因此很多专家学者对参数估计问题做了大量的工作,提出了矩估计、极大似然估计和贝叶斯估计等常用的估计方法。大样本情况下,矩估计、极大似然估计和贝叶斯估计均可以获得准确的结果。小样本情况下,通常使用贝叶斯估计,但贝叶斯估计的计算中通常涉及复杂的二重积分,无法得到显式解。本文从参数的联合共轭先验是正态-倒伽玛分布的贝叶斯估计结果获得启发,在统计量中加入样本均值的平方信息来改进贝叶斯估计,提出了依赖于X,X2和S2这三个统计量的线性结构的贝叶斯估计。本文首先计算出了基于X,X2和S2的三个统计量和基于X和S2两个统计量的线性结构贝叶斯估计的表达式,其次从理论上证明了在均方误差矩阵准则下:基于三个统计量的线性结构贝叶斯估计要优于基于两个统计量下的线性结构贝叶斯估计以及无偏估计和极大似然估计。在数值模拟比较时,我们结合MCMC方法获得了贝叶斯估计,同时考察了Lindley 贝叶斯近似计算方法。先验取μ|σ2～N(μ0,σ2/k0),σ2～IGa(v0/2,v0σ02/2)时,我们发现使用X,X2和S2这三个统计量的线性结构贝叶斯估计与使用通常的贝叶斯估计得到参数μ的估计结果一致,且参数σ2的线性结构贝叶斯估计与其贝叶斯估计非常接近。对于用Lindley近似计算获得的贝叶斯估计来说,在不同的先验分布下,有时优于线性结构贝叶斯估计,有时又劣于线性结构贝叶斯估计,此时要根据不同的先验进行具体的分析。但对于不同统计量的线性结构贝叶斯估计,随着样本容量的增加,线性结构贝叶斯估计总是向贝叶斯估计趋近,最终相对误差函数δ的图像稳定于函数y1=n-2和y2=n-1的图像之间。
[Abstract]:Normal distribution plays a very important role in the theoretical research and practical application of mathematical statistics. More and more research fields are involved in the parameter estimation of normal distribution. Therefore, many experts and scholars have done a lot of work on the parameter estimation problem, and put forward the common use of moment estimation, maximum likelihood estimation and Bias estimation. Estimation method. In large sample case, moment estimation, maximum likelihood estimation and Bias estimation can obtain accurate results. Under small sample cases, the Bias estimate is usually used, but the calculation of Bias's calculation usually involves complex double integral and cannot get explicit solution. The joint conjugate prior of the parameters is normal - inverted gamma. The Bayesian estimation results of the distribution are inspired. The Bayesian estimation of the three statistics dependent on the three statistics of X, X2 and S2 is proposed by adding the square information of the sample mean to the statistics. This paper first calculates the linear structure of the three statistics based on X, X2 and S2 and based on the two statistics of X and S2. Bias's estimated expression, then theoretically proved that under the mean square error matrix principle: the linear structure Bias estimation based on three statistics is better than the linear structure Bias estimation based on two statistics, unbiased estimation and maximum likelihood estimation. In numerical simulation comparison, we combine the MCMC method to obtain the shells. Juliu estimated that, at the same time, the Lindley Bayesian approximation calculation method was also examined. We found that the linear structural Bayesian estimation using the three statistics of X, X2 and S2 (v0/2, V0 Sigma 02/2) using the three statistics of X, X2 and S2 (v0/2, V0 Sigma 02/2) was found to be consistent with the estimation results obtained by the usual Bayesian estimation and the linear structural shellfish of parameter sigma 2. Juliu estimation is very close to its Bayesian estimation. For Bayesian estimation obtained by Lindley approximation, it is sometimes better than linear Bayesian estimation under different prior distributions, and sometimes worse than linear Bayesian estimation. At this time, specific analysis should be carried out according to different priors. The Bayesian estimation of sexual structure, with the increase of sample size, the Bayesian estimation of linear structure always converge to the Bayesian estimation, and the final relative error function [delta] image is stable between the images of the function y1=n-2 and the y2=n-1.
【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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