Bayesian methods are effective statistical one that combine prior information with samples, but the inference results are different because subjectivity is involved in representing prior information as a prior distribution. The empirical Bayes approach is a statistical method that performs continuous Bayesian decision making when the prior distribution is unknown but given a large number of past data, which overcomes the above drawback and is now a common way because of a vast amount of data available. The inference for truncation parameters is important in evaluating the lower or upper bounds of the population and is required in many fields such as reliability, meteorology, medicine, etc. Many authors have considered the case of one-sided truncated distribution in the empirical Bayes framework, but in practice we are faced with the case of two-sided one. We consider the empirical Bayes estimation for truncation parameters of two-sided truncated distribution under the squared error loss. First, Bayesian estimators of the truncation parameters are derived to minimize the Bayes risk using sample of size two. And, based on the Bayesian estimators and kernel density estimate of the population density function, empirical Bayes estimators of the truncation parameters are constructed. Next, under some assumptions asymptotic optimality of empirical Bayes estimators is proved when the sample size goes to infinity, and the convergence rate is also evaluated. It is also proved that the probability that the lower and upper bounds are reversely estimated approaches zero. Finally, an example is presented to show the validity of the assumptions and the performance of the proposed empirical Bayes estimators is evaluated through simulation on the example. New proposal of using size two samples could be extended to the case of multi-dimensional truncated distributions defined on hyper-cubic domain.
Ri et al. (Thu,) studied this question.