Inadmissibility Results for the Selected Hazard Rates

Let us consider $k ~(\ge 2)$ independent populations $\Pi_1, \ldots,\Pi_k$, where $\Pi_i$ follows exponential distribution with hazard rate ${\sigma_i},$ ($i = 1,\ldots,k$). Suppose $Y_{i1},\ldots, Y_{in}$ be a random sample of size $n$ drawn from the $i$th population $\Pi_i$, where $i = 1,\ldots,k.$ For $i = 1,\ldots,k$, consider $Y_i=\sum_{j=1}^nY_{ij}$. The natural selection rule is to select a population associated with the largest sample mean. That is, $\Pi_i$, ($i = 1,\ldots,k$) is selected if $Y_i=\max(Y_1,\ldots,Y_k)$. Based on this selection rule, a population is chosen. Then, we consider the estimation of the hazard rate of the selected population with respect to the entropy loss function. Some natural estimators are proposed. The minimaxity of a natural estimator is established. Improved estimators improving upon the natural estimators are derived. Finally, numerical study is carried out in order to compare the proposed estimators in terms of the risk values.