Empirical Bayes Estimation And Test For Continuous Two-side Truncated Distribution Families Under α-Mixing Samples

The Bayesin method was introuduced in 1763,and the Empirical Bayes approach was put forward by Robbins in 1955.Empirical Bayes is a successful combination of Bayes and frequen-tist,it inherits Bayes’ thought that consider the parameter as a random variable,takes the prior information into account,and uses frequentist’s methods to obtain the information about the prior distribution.Construct a suitable estimate is the first and primary point of the Empirical Bayes estimation.In this paper,we discussed the Empirical Bayes estimation and test for continuous two-side truncated distribution families under α-mixing samples.On one hand we get the Empirical Bayes estimation of the parameters under certain loss function,derive the Empirical bayes estimation of the test function under another loss function.On the other hand,we prove the proposed estimators and test are the asymptotically optimal estimator and the asymptotically optimal test function.Then we get the convergence rates of them.What’s more,an example that satisfies the conditions in the main results is given and then we conduct some simulated research.Here we summary some new findings in this paper:1.In this paper,we discuss the Empirical Bayes estimation and test for continuous two-side truncated distribution families under the assumption that the current samples and historical sam-ples for a whole α-mixing sample.The research under independent identically distributed samples becoming more and more perfect,but in dependent samples,litter progress was made in this aspect before.Even though the study of many papers made efforts under dependent samples,but most of them assumed that the current samples and historical samples are independent.Therefore it is necessary to study Empirical Bayes method under the assumption that the current samples and historical samples for a whole is a dependent sample.2.In this paper,we not only get the the Empirical Bayes estimation for continuous two-side truncated distribution families under α-mixing samples,prove that the estimator is the asymptoti-cally optimal one,and obtain its convergence rate,we also consider a test function of the proposed distribution families,and prove the same qualities.3.In this paper,the convergence rate of parameter estimation and test are simulated.Through the numerical simulation,we find that:for both of the estimation and test,with the increasing of sample size,the difference between the risk of empirical bayes and the risk of bayes is becoming smaller and smaller.