Linear Structural Bayes Estimator For Parameters Of Normal Distribution

The normal distribution occupy a very important position in the theory and the application of mathematical statistics.Many experts and scholars have done a lot of research in parameter estimation which is the core of the research in normal distribution,such as:the moment estimation,the maximum likelihood estimation and the Bayes estimator.Among them,the moment estimation,the maximum likelihood estimation and the Bayes estimator can obtain accurate results usually in the case of large samples.Meanwhile,in case of small samples,we normally use Bayes estimator method.But in the calculation we often cannot get accurate results due to the complex integrals.This article enlights from the specific expression of the Bayes estimator with the prior distribution being ?|?2?N(?0,?2/k0)and ?2?IGa(v0/2,v0?02/2),considers to improve the Bayes estimator by adding square information of sample mean in statistics,the paper propose the linear structural Bayes estimator for parameters of normal distribution based on X,X2 and S2Frist,this paper calculates the expressions of the linear structural Bayes estimator based on the two statistics and the three statistics.Then,under the mean square error matrix criterion,we prove that the linear structural Bayes estimator based on three statistics is better than that based on two statistics,the unbiased estimation and the maximum likelihood estimation as well.We use the MCMC method to obtain the Bayes estimator to exhibit the numerical simulate comparison,and use the Lindley approximation to investigate the estimator of Bayes approximate calculation for addition.The estimation of ? is the same as the usual Bayes estimator and the estimation of ?2 using the linear structural Bayes estimator based on three statistics is very close to the usual Bayes estimator in the prior distribution with ?|?2?N(?0,?2/k0)and ?2?IGa(v0/2,v0?02/2).We get different results with different priors by using Lindley approximation,sometimes it is better than the linear structural Bayes estimator,but sometimes it is inferior to the linear structural Bayes estimator,we should carry out specific analysis according to different priors.In summary,the linear structural Bayes estimator finally approaches to the Bayes estimator as the sample size increases.The curve of relative error function ? stably stays between the functions yl=n2 and y2=n-1.