Improved estimation of the Poisson parameter

Abstract

A preliminary test approach using shrinkage technique is proposed for the estimation of a Poisson parameter. In this article, an a priori value of the parameter is assumed to be available in the form of realistic guessed value based on the experimenter's knowledge and experience to increase the precision of estimators by using a preliminary test. It is to be noted that this approach differs from the Bayesian approach since we do not assume a prior distribution for the parameter. Three possible estimators, namely the unrestricted maximum likelihood estimator (UMLE), the shrinkage restricted maximum likelihood estimator (SRMLE) and the shrinkage preliminary test maximum likelihood estimator (SPTMLE), are presented. Asymptotic mean squared errors of the estimators are derived and compared analytically and numerically. The relative dominance picture of the estimators is presented. It is shown that the range in the parameter space which SPTMLE dominates the UMLE is wider than that of usual preliminary test maximum likelihood estimator (PTMLE). Further, the SPTMLE provides more meaningful size for the preliminary test than the usual PTMLE. A Monte Carlo study provided to compare the performance of the estimators for small sample sizes.