Admissibility of Linear Estimators in the One Parameter Exponential Family

Abstract

For estimating the mean in the one parameter exponential family with quadratic loss, Karlin (1958) gave sufficient conditions for admissibility of estimators of the form $aX$. Later, Ping (1964) and Gupta (1966) gave sufficient conditions for admissibility of estimators of the form $aX + b$ for the same problem. Zidek (1970) gave sufficient conditions for the admissibility of $X$ for estimating an arbitrary piecewise continuous function of the parameter, say $\gamma(\theta)$, not necessarily the mean. In this paper it is shown that Karlin's argument yields sufficient conditions for the admissibility of estimators of the form $aX + b$ for estimating $\gamma(\theta)$. The results are then extended to the case when the parameter space is truncated.