Equivariant Estimation of the Selected Guarantee Time

TL;DR

This paper develops equivariant estimators for the location parameter of the selected exponential population, based on a natural selection rule, and analyzes their optimality, admissibility, and minimax properties through theoretical results and simulations.

Contribution

It introduces new equivariant estimators for selected population parameters and establishes their optimality and admissibility properties in the exponential model.

Findings

UMVUE derived for the selected best population

Generalized Bayes estimator identified as BAEEs

Simulation results compare estimator performances

Abstract

Consider two independent exponential populations having different unknown location parameters and a common unknown scale parameter. Call the population associated with the larger location parameter as the "best" population and the population associated with the smaller location parameter as the "worst" population. For the goal of selecting the best (worst) population a natural selection rule, that has many optimum properties, is the one which selects the population corresponding to the larger (smaller) minimal sufficient statistic. In this article, we consider the problem of estimating the location parameter of the population selected using this natural selection rule. For estimating the location parameter of the selected best population, we derive the uniformly minimum variance unbiased estimator (UMVUE) and show that the analogue of the best affine equivariant estimators (BAEEs) of location parameters is a generalized Bayes estimator. We provide some admissibility and minimaxity results for estimators in the class of linear, affine and permutation equivariant estimators, under the criterion of scaled mean squared error. We also derive a sufficient condition for inadmissibility of an arbitrary affine and permutation equivariant estimator. We provide similar results for the problem of estimating the location parameter of the selected population when the selection goal is that of selecting the worst exponential population. Finally, we provide a simulation study to compare, numerically, the performances of some of the proposed estimators.