The Small-Sample Variance of the Pitman Location Estimators

Abstract

For three symmetric distributions and six sample sizes, this article presents estimates of the small-sample variance of Pitman's location-invariant and location-scale-invariant estimators of location. It then compares these two estimators and investigates the closeness of the Cramér-Rao bound when estimators are required to be invariant. Expressions of the form c/(n − d) prove quite effective in fitting variances of Pitman estimators, and d can be interpreted as the amount of Fisher information “lost.” In terms of relative efficiency, maximum-likelihood estimators and linear combinations of order statistics offer computationally attractive alternatives to the Pitman estimators.