On Minimum Variance in Nonregular Estimation

Abstract

A case of nonregular estimation arises in attempting to estimate a single unknown parameter, $\theta$, in the probability distribution of a single chance variable in which one or both of the extremities of the range of the distribution are functions of the unknown parameter. The case treated in this paper is the one in which a probability density of exponential type exists. When one extremity alone of the range depends non-trivially upon $\theta$, a necessary and sufficient condition is given in order that a single order statistic be a sufficient statistic for $\theta$. In this case conditions are given for the existence of a unique unbiased estimate of $\theta$ possessing minimum variance uniformly in $\theta$. In the case in which both extremities of the range depend upon $\theta$, a necessary and sufficient condition is given that the smallest and largest order statistics constitute a set of sufficient statistics for $\theta$. In this case Pitman [1] has shown that a single sufficient statistic exists if one extremity of the range is a monotone decreasing function of the other extremity. It is shown that under the above condition a unique unbiased estimate exists possessing minimum variance. Moreover a surmise of Pitman is proved that only under this condition does a single sufficient statistic exist. When a single sufficient statistic does not exist, an unbiased estimate of a known function of $\theta$ is obtained which has less variance than any analytic function of the set of sufficient statistics for $\theta$.