Estimating average worth of the selected subset from two-parameter exponential populations

Abstract

Suppose independent random samples are available from k ( k ≥ 2) exponential populations ∠1 , ⋯ , ∠k with a common location θ and scale parameters σ1 , ⋯ , σk, respectively. Let Xi and Yi denote the minimum and the mean, respectively, of the i th sample, and further let X = min{X1 , ⋯ , Xk} and Ti = Yi - X ; i = 1, ⋯ , k . For selecting a nonempty subset of {∠1 , ⋯ ,∠k} containing the best population (the one associated with max{σ1 , ⋯ , σk}), we use the decision rule which selects ∠i if Ti ≥ c max{T1, ⋯ , Tk}, i = 1, ⋯ , k. Here 0 < c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem is to estimate the average worth W of the selected subset, the arithmetic average of means of selected populations. In this article, we derive the uniformly minimum variance unbiased estimator (UMVUE) of W. The bias and risk function of the UMVUE are compared numerically with those of analogs of the best affine equivariant estimator (BAEE) and the maximum likelihood estimator (MLE).