Alternative Estimators for the Scale Parameter of the Exponential Distribution with Unknown Location

Abstract

Let $X_1, X_2,\cdots, X_n$ be independent observations from an exponential distribution with an unknown location-scale parameter $(\mu, \sigma)$. Let $\bar{X} = n^{-1} \sum X_i$ and $M = \min X_i$. Under squared error loss the best location-scale equivariant estimator of $\sigma$ is $\bar{X} - M$, which agrees with the maximum likelihood estimator. Arnold (J. Amer. Statist. Assoc. 65 (1970) 1260-1264) and Zidek (Ann. Statist. 1 (1973) 264-278) have shown that $\bar{X} - M$ is inadmissible but the dominating estimator which they produce is probably inadmissible as well. In this paper a "smoother" dominating procedure is presented, and the risk functions of the various alternatives are plotted. Similar results are obtained for strictly bowl-shaped loss functions.