Several Distribution Parameters Of Bayesian Statistical Inference And Applications

In this paper, Bayesian and empirical Bayesian theory are used to study the statistical analysis for Γ distribution, normal distribution, scale parameter family and the failure function in reliability tests.Firstly, under the conjugate prior distribution, we obtain the Bayes estimators of the loss function and risk function for Γ parameter estimator and normal variance estimator, also discuss the conservative property and the rationality of these Bayes estimators. Then the similar results are extended to the scale parameter family.Secondly, under the square loss, we construct the asymptotically optimal and admissible empirical Bayes estimator of the mean for normal distribution, theconvergence rate O(n^-1) is also obtained. In the end, an illustrative example isexamined numerically by means of the Monte-Carlo simulation. It can be shown that the presented EB estimator is very close to the real value.Thirdly, the simple forms of the minimum risk equivalent (MRE) estimators for a class of scale parameter are derived, and we study the U-admissibility of two class of equivalent estimators. Specially, we discuss the U-admissibility of several MRE estimators under different loss functions.Finally, we investigate the empirical Bayes test problem of the failure function based on type II censored samples in the exponential distribution. The asymptotically optimal EB test function has been constructed by kernel estimation method and the convergence rates are established.