Inadmissibility Results for the Selected Scale Parameters

Abstract

Let $X_1, X_2, \ldots, X_k$ be $k$ independent gamma random variables with different scale parameters but with a common known shape parameter. Suppose the population corresponding to the largest $X_{(1)}$ [or the smallest $X_{(k)}$] observation is selected. The problem of estimating the scale parameter $\theta_{(1)}$ [or $\theta_{(k)}$] of the selected population is considered. We derive, using the method of differential inequalities, explicit estimators that dominate the natural or the existing estimators. The improved estimators of $\theta_{(1)}$ are similar to that of DasGupta estimators for the usual simultaneous estimation problem. An implication of this result for the simultaneous estimation of the selected subset is also considered.