Convergence rates for empirical Bayes two-action problems: the uniform U(0,θ) distribution

Mohamed Tahir

doi:10.1155/s1048953392000145

Abstract

The purpose of this paper is to study the convergence rates of a sequence of empirical Bayes decision rules for the two-action problems in which the observations are uniformly distributed over the interval (0,θ), where θ is a value of a random variable having an unknown prior distribution. It is shown that the proposed empirical Bayes decision rules are asymptotically optimal and that the order of associated convergence rates is O(n−α), for some constant α, 0<α<1, where n is the number of accumulated past observations at hand.

Highlights

In situations involving sequences of similar but independent statistical decision problems, it is reasonable to formulate the component problem in the sequence as a Bayes statistical decision problem with respect to an unknown prior distribution over the parameter space, and use the accumulated observations from the previous decision .problems to improve the decision rule at each stage
Robbins [6] and Samuel [7] exhibit empirical Bayes rules for the two-action problems in which the distributions of the observations belong to a certain exponential family of probability distributions
The objective of this paper is to investigate the convergence rate of a sequence of empirical Bayes decision rules for the two-action problems in which the observations are uniformly distributed over the interval (0,), where is a value of a random variable having an unknown prior distribution

Summary

INTRODUCTION

In situations involving sequences of similar but independent statistical decision problems, it is reasonable to formulate the component problem in the sequence as a Bayes statistical decision problem with respect to an unknown prior distribution over the parameter space, and use the accumulated observations from the previous decision .problems to improve the decision rule at each stage. Johns and Van l:tyzin [1] study the convergence rates of a sequence of empirical Bayes rules which they propose for the two-action problems where the observations are members of some continuous exponential families of distributions. The objective of this paper is to investigate the convergence rate of a sequence of empirical Bayes decision rules for the two-action problems in which the observations are uniformly distributed over the interval (0,), where is a value of a random variable having an unknown prior distribution. The Bayes risk incurred by Convergence Rates for Empirical Bayes Two.Action Problems using the decision rule 8 with respect to the prior distribution G is given by b.

THE PROPOSED EMPIRICAL BAYES RULES

ASYMPTOTIC OPTIMALITY

EXAMPLE