Estimating a parametric function involving several exponential populations

Abstract

This article provides some optimal estimators for a parametric function θR, which arises in the study of reliability analysis involving several exponential populations. Let (Formula presented.) be (Formula presented.) 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 (Formula presented.) be the reliability function of the ith system, and let (Formula presented.) denote the largest value of (Formula presented.) ’s at a fixed t. We call the system associated with (Formula presented.) 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 (Formula presented.) and the random variables. The uniformly minimum variance unbiased estimator (UMVUE) and the generalized Bayes estimator of θR are derived. Two natural estimators (Formula presented.) and (Formula presented.) of θR are also considered. A general result for improving an equivariant estimator of θR is derived. Further, we show that the natural estimator (Formula presented.) 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. © 2022 Taylor & Francis Group, LLC.