Empirical Bayes with a changing prior
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis examines a modified empirical Bayes decision problem in which the a priori distribution is assumed to change with each successive component problem. Firstly, the usual empirical Bayes problem and associated rates of convergence to optimality is considered and rates are given for certain parametric examples. General results on asymptotic optimality for the modified problem are given. The problem is then examined for location parameter families in which the a priori mean is assumed to change in a linear fashion. Asymptotically optimal estimators are given for both the two action problem and the estimation problem under squared error loss. Generalised convergence rates are developed for the exponential family of densities. Some examples of parametric a priori distributions are also considered. The problem of the selection of the 'best' of several populations is studied in the modified empirical Bayes framework. Asymptotically optimal procedures for this problem, under a linear loss structure, are developed. The modified empirical Bayes method is applied to a system reliability model. Finally the modified problem is extended to examine the case where the a priori means are random variables, generated by a martingale process. Asymptotically optimal estimators, with associated rates of convergence, are established here under the assumption that the a priori distributions are members of a parametric family.