Estimation After Selection From Uniform Populations With Unequal Sample Sizes

Abstract

SYNOPTIC ABSTRACTConsider k (⩾ 2) independent uniform populations π1, …, πk, where πi ≡ U(0, θi), and θi > 0 (i = 1, …, k) is an unknown scale parameter. For selecting the unknown population having the largest scale parameter, we consider a class of selection rules based on the natural estimators of θi, i = 1, …, k. We consider the problem of estimating the scale parameter θS of the selected population, using a fixed selection rule from this class, under the scaled-squared error loss function. We derive the uniformly minimum variance unbiased estimator (UMVUE) of θS. We also consider three natural estimators ϕN, 1, ϕN, 2, and ϕN, 3 of θS which are, respectively, based on the maximum likelihood estimators, UMVUEs, and minimum risk equivariant estimators for component estimation problems. The natural estimator ϕN, 3 is shown to be a generalized Bayes estimator with respect to a non informative prior. Further, we derive a general result for improving a scale-invariant estimator of θS. Using this result, the estimators better than the UMVUE and the natural estimator ϕN, 1 are obtained. It is also shown that a subclass of natural-type estimators, which contains the natural estimator ϕN, 2, is inadmissible for estimating θS under the scaled-squared error loss function. Finally, we provide a simulation study on the performances of various competing estimators of θS.