Efficient Estimation of a Shift Parameter From Grouped Data

Abstract

Universally efficient procedures for testing and estimation problems have been briefly explored by Hajek [3] and Stein [7]. In this paper we consider two populations having frequency functions $f(x)$ and $f(x - \theta)$ where the common form $f$ and the shift parameter $\theta$ are unknown. A method of estimating $\theta$ when one sample is reduced to a frequency distribution over a given set of class-intervals is suggested by the likelihood principle and the asymptotic efficiency of this estimator relative to the appropriate maximum likelihood estimator based on the complete data is found to be the ratio of the Fisher-information in a grouped observation to the Fisher-information in an ungrouped observation.