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.