Minimum phi-divergence estimators of a set of binomial probabilities

Abstract

In this paper we consider the product Bernoulli model with different probability success and the related problem of estimation of the probabilities when it is suspected that they are equal. If we are completely sure that the probabilities are equal we must use a restricted estimator but in many situations it is not clear if the probabilities are equal or not and then a better procedure will be to use a "preliminary test estimator". Based on minimum phi-divergence estimator (MφE) we study, in this paper, some estimators for the parameters of the product Bernoulli model: Unrestricted MφE, Restricted MφE, Preliminary MφE, Shrinkage MφE, Shrinkage preliminary MφE, James-Stein MφE, Positive-part of Stein-Rule MφE and Modified preliminary MφE. Asymptotic quadratic bias as well as asymptotic quadratic risk are studied under contiguous alternative hypotheses. © 2011 Springer-Verlag Berlin Heidelberg.