On Emprical Bayes Estimation of Nonidentical Components

• Activity Number: 444
• Type: Contributed
• Date/Time: Thursday, August 7, 2003 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Stat. Sciences
• Author(s): Rohana J. Karunamuni*+
• Companies: University of Alberta
• Keywords: Empirical Bayes ; Exponential Family ; Asymptotic Optimality

Abstract

In this paper, we consider empirical Bayes squared loss estimation problem in the nonidentical components case for the continuous one-parameter exponential family. That is, the i^{th} component is based on the pair (X_{i}, \theta _{i}), where, given \theta _{i}, X_{i} is distributed according to density of the form f_{i}(x\mid \theta _{i})=(h(\theta _{i}))^{n_{i}}e^{x\theta _{i}}l_{n_{i}}(x) for x\geq a and \theta _{i}\thicksim G, i\geq 1, with G unknown. An improved empirical Bayes of the "present'' component parameter \theta _{m+1} is constructed using the "past'' data X_{1}, X_{2}, ..., X_{m} as well as the present data X_{m+1}. Asymptotic optimality of the proposed estimator along with a rate of convergence of the corresponding regret are established. It is observed that the nonidentical component case raises interesting and difficult problems in regard to the efficient use of data implementation. The method is used to obtain estimates of log odds ratio of two datasets.