Inference on Selected Population under Generalized Stein Loss Function

Abstract

Inference on selected population is concerned with the problem of selecting the best population among the given k populations, and then doing inference on the parameter of selected population. Suppose independent random samples (Xi1,…,Xin), i=1,…,k are drawn from U(0,ϴi) population, respectively. Let Xi= max(Xi1,…,Xin) and X(1)≤X(2)≤…≤X(k) be the order statistics of X1,…,Xk. The population corresponding to largest X(k) (or the smallest X(1)) is selected and the problem of estimation the parameter ϴM (or ϴJ) of the selected population under generalized Stein loss function is considered. We obtain the Uniformly Minimum Risk Unbiased (UMRU) estimator of ϴM (and ϴJ) and show that the UMRU estimator of ϴM is inadmissible. For k=2, we derive the class of all linear admissible estimators of ϴM  and ϴJ, respectively.