The Bayes Estimation And Characters Of Several Distribution Parameters Under Entropy Loss Function

With the advances in science and technology and the development of the society, the Bayesian statistics as a new kind of statistics method is developed gradually, it is constituted the important part of probability and statistics with the classical statistical. Compared with the classical statistics method, due to the Bayesian statistics method can synthesize all kinds of information to statistics inference as much as possible, and also can more effectively combine theory with the actual situation, namely considering the consequences and benefits, the Bayesian statistics method play a more and more important role in all aspects of actual life. To introduce the loss function is to consider the consequence, as an important measure of statistical decision, loss function plays a very important role in statistical inference. Parameter estimation as one of the main problems of statistical inference, and the related properties of parameter estimation can analysis to the admissibility of estimation from the perspective of statistical decision, which helps us to make correct and reasonable statistical inference. Because of Exponentiated-Weibull(EW) distribution, Log-normal(LN) distribution and Pareto distribution are commonly used distribution is widely used in engineering, reliability and economics, this article are mainly studied these three distributions are under asymmetry Entropy loss function, and the corresponding practical application and the instance simulation are presented in this paper.Firstly, the research background, purpose and significance of this article are mainly introduced, and the development of Bayesian theory is introduced, which points out the limitation of the classical statistical method and the excellent properties of the Bayesian statistics method by using the prior information, sample information and general information to estimate parameter. Then the basic theory of the Bayesian estimation method and Empirical Bayesian estimation method are introduced, and points out the lack of symmetry loss function at the same time. The present research situation of asymmetric Entropy loss function is introduced, and the method of converting the Entropy loss function to LINEX loss function according to the relationship between the LINEX loss function and Entropy loss function and the concept that then using the existing conclusions to solve the problem.Secondly, under the Entropy loss function, the Bayesian estimation of parameter for Exponentiated-Weibull distribution, Log-normal distribution and Pareto distribution under the conjugate prior distribution, Jeffrey’s prior distribution and multi-layer prior distribution are discussed when the prior distribution is known, the simulation studies are given for using the Monte Carlo method. The Empirical Bayesian estimation of these three distributions is constructed by using the kernel density estimation when the prior distribution is unknown. The Bayesian estimation of asymptotically optimality is proved by using the inequality() 2() 1()a b a be a b e e---- ≤-(including a >0,b >0), Holder inequality and Markov inequality, and the Bayesian estimation of admissibility is proved by reduction to absurdity. The specific expression of Bayesian estimation for Pareto distribution parameters are given according to the Lindley approximation theorem when the two parameters are all unknown.Finally, the reliability R and failure rate λ for Exponentiated-Weibull distribution are regarded as random variables or a function about θ parameter, the posterior density of reliability R and failure rate λ are solved by using the related properties of density function. Then the Maximum likelihood estimation and the Bayesian estimation of reliability R and failure rate λ for Exponetiated-Weibull distribution under the Entropy loss function are studied, and the simulation studies are given for using the Monte Carlo method. The results show that the Bayesian estimation under the Entropy loss function is better than the square loss function and the Maximum likelihood estimation.Then the Bayesian estimation for Product’s Radiation Hardening Performance by using Bayesian estimation method, and using the example to prove the rationality of the evaluations method, which is put forward in this paper.