Reliability estimation of the selected exponential populations

Let Π1,Π2,…,ΠkΠ1,Π2,…,Πk be kk populations with ΠiΠi being exponential with an unknown location parameter μiμi and a common but known scale parameter σσ, i=1,…,ki=1,…,k. Suppose independent random samples are drawn from the populations Π1,Π2,…,ΠkΠ1,Π2,…,Πk. Let {Xi1,Xi2,…,Xin}{Xi1,Xi2,…,Xin} denote the sample drawn from iith population, i=1,…,ki=1,…,k. A subset of the populations with high reliabilities is selected according to Gupta’s [Gupta, S.S., 1965. On some multiple decision (Selection and Ranking) rules. Technometrics 7, 225–245] subset selection procedure. We consider the problem of estimating simultaneously the reliability functions of the populations in the selected subset. The uniformly minimum variance unbiased estimator (UMVUE) is derived and its inadmissibility is established. An estimator improving the natural estimator is also obtained by using the differential inequality approach used by Vellaisamy and Punnen [Vellaisamy, P., Punnen, A.P., 2002. Improved estimators for the selected location parameters. Statist. Papers 43, 291–299].