On estimating the location parameter of the selected exponential population under the LINEX loss function

Abstract

Suppose that $\pi_{1},\pi_{2},\ldots ,\pi_{k}$ be $k(\geq2)$ independent exponential populations having unknown location parameters $\mu_{1},\mu_{2},\ldots,\mu_{k}$ and known scale parameters $\sigma_{1},\ldots,\sigma_{k}$. Let $\mu_{[k]}=\max \{\mu_{1},\ldots,\mu_{k}\}$. For selecting the population associated with $\mu_{[k]}$, a class of selection rules (proposed by Arshad and Misra [Statistical Papers 57 (2016) 605–621]) is considered. We consider the problem of estimating the location parameter $\mu_{S}$ of the selected population under the criterion of the LINEX loss function. We consider three natural estimators $\delta_{N,1},\delta_{N,2}$ and $\delta_{N,3}$ of $\mu_{S}$, based on the maximum likelihood estimators, uniformly minimum variance unbiased estimator (UMVUE) and minimum risk equivariant estimator (MREE) of $\mu_{i}$’s, respectively. The uniformly minimum risk unbiased estimator (UMRUE) and the generalized Bayes estimator of $\mu_{S}$ are derived. Under the LINEX loss function, a general result for improving a location-equivariant estimator of $\mu_{S}$ is derived. Using this result, estimator better than the natural estimator $\delta_{N,1}$ is obtained. We also shown that the estimator $\delta_{N,1}$ is dominated by the natural estimator $\delta_{N,3}$. Finally, we perform a simulation study to evaluate and compare risk functions among various competing estimators of $\mu_{S}$.