Statistical Inference Research Based On The Bayes Estimation Of The Parameters Of Burr X Distribution

Until now the development of parameter estimation, normal conventional estimation methods are still far beyond the demands of precision in many conditions. Since Bayesian estimation method has been introduced by T. R. Bayes, it was received a popular concern due to its satisfactory estimation characteristics. With the development of200years, the Bayesian estimation method finally developed into a systematic subject.This paper firstly considered the Bayesian estimation of character based on Burr X distribution of single parameter. Thanks to exponential family of distributions and the nature of complete sufficient statistics, we work out the uniformly minimum variance unbiased estimation (UMVU estimate) of parameter, meanwhile we get its entropy loss function. Then when we set the prior distribution as X distribution, we can finally find out four kinds of Bayesian estimations according to Burr X distributions and discuss about asymptotic optimality of empirical Bayesian estimations. Otherwise armed with the data simulation technology, we do a simulation of asymptotic optimality of empirical Bayesian estimation, comparing robustness of Hierarchical Bayes estimation method based on super prior distribution and E-Bayes estimation that proves out Hierarchical Bayes estimation method behaves better than E-Bayes. Comparing mean square error of those several estimation methods, it turns out that the least one is empirical Bayesian estimation method, then follows with uniformly minimum variance unbiased estimation, Hierarchical Bayes estimation method and then the last one is E-Bayes estimation. Actually, it is rather hard to determine the prior distribution when we use Bayesian estimation method. In order to evaluate various effects of prior distribution on Bayesian estimations, we consider the robustness of Bayesian estimation based on conjugate prior distribution and ε-pollution distribution. Through the data simulation, we find out conjugate prior distribution does a better job behaving a better robustness. However under the kinds of empirical distributions, the variations of samples really have strong influence on robustness. Besides we also discuss about maximum likelihood estimation and Bayesian estimation of parameters based on Burr X distribution of two parameters. Due to its complicated form, we forward a kind of iterate method and two kinds of relative methods (Lindley’s approximate and MCMC) to solve out numerical solution. Making a compare with a group of simulated data and a group of actual data, it turns out that Lindley’s approximate has the highest accuracy, then comes to Bayesian estimation, and maximum likelihood estimation.The paper makes a conclusion of current research circumstance of Burr X distribution. Based on the former outcome of Bayesian estimation, we discuss several kinds of Bayesian estimation methods, at the same time, we make a compare between with several kinds of methods and classical estimation method. After a numerical simulation analysis, we find out Bayesian estimation method has a high accuracy.