On location parameter family of distributions with uniformly minimum variance unbiased estimator of location

Abstract

Suppose that XI, ..., X n are real random variables distributed according to a continuous distribution with a location parameter. Let the density functions of X i be denoted by f(x-e). The purpose of this paper is to prove that under some set of regularity conditions, uniformly minimum variance unbiased (UMVU) estimator exists if and only if the distribution is either normal or expo-gamma type in one case and exponential in another. The proof of the main theorems is done in three steps. First it is shown that if UMVU estimator exists, it is equivalent to the Pitman or the best location invariant estimator. Secondly, it will be shown that the Pitman estimator is UMVU only if it is a sufficient statistic. Finally, under a set of regularity conditions it is shown that the location parameter family admits one-dimensional sufficient statistic if and only if the distribution is either normal or expo-gamma type, or in another situation, exponential.