Estimation after selection from exponential populations with unequal scale parameters

Abstract

Consider $k$ ($\ge$2) exponential populations having unknown guarantee times and known (possibly unequal) failure rates. For selecting the unknown population having the largest guarantee time, with samples of (possibly) unequal sizes from the $k$ populations, we consider a class of selection rules based on natural estimators of the guarantee times. We deal with the problem of estimating the guarantee time of the population selected, using a fixed selection rule from this class, under the squared error loss function. The uniformly minimum variance unbiased estimator (UMVUE) is derived. We also consider two other natural estimators $\varphi _{N,1}$ and $\varphi _{N,2}$ which are, respectively, based on the maximum likelihood estimators and UMVUEs for component problems. The estimator $\varphi _{N,2}$ is shown to be generalized Bayes. We further show that the UMVUE and the natural estimator $\varphi _{N,1}$ are inadmissible and are dominated by the natural estimator $\varphi _{N,2}$. A simulation study on the performance of various estimators is also reported.