A Class of Nonlinear Admissible Estimators in the One-Parameter Exponential Family

Abstract

We are concerned with the admissibility of nonlinear estimators of the form $(aX + b)/(cX + d)$ in the one-parameter exponential family, in estimating $g(\theta)$ with quadratic loss. Our method will be reminiscent of that of Karlin who gave sufficient conditions for admissibility of linear estimators $aX$ in estimating the mean in the one-parameter family. Our results generalize those of Ghosh and Meeden who studied admissibility of $aX + b$ for estimating an arbitrary function $g(\theta)$. Particular cases of estimators of the form, $c/X$ are studied and several examples are given. We show that $(n - 2)/(X + a), a \geq 0$ is admissible in estimating the parameter of an exponential density. We also discuss the case of truncated parameter space.