Estimating a parametric function involving several exponential populations

This article provides some optimal estimators for a parametric function θR, which arises in the study of reliability analysis involving several exponential populations. Let π1,π2,…,πk be k (≥2) independent populations, where the population πi follows an exponential distribution with unknown guarantee time and a known failure rate. These populations may represent the lifetimes of k systems. Let θi(t) be the reliability function of the ith system, and let θ(k) denote the largest value of θi(t)’s at a fixed t. We call the system associated with θ(k) the best system. For selecting the best system, a class of natural selection rules is used. The goal is to estimate the parametric function θR, which is a function of parameters θ1,θ2,…,θk, and the random variables. The uniformly minimum variance unbiased estimator (UMVUE) and the generalized Bayes estimator of θR are derived. Two natural estimators δN,1 and δN,2 of θR are also considered. A general result for improving an equivariant estimator of θR is derived. Further, we show that the natural estimator δN,2 dominates the UMVUE under the squared error loss function. Finally, the risk functions of the various competing estimators of θR are compared via a simulation study.