Estimation of the parameter of the selected uniform population under the entropy loss function

Suppose independent random samples Xi1,…,XinXi1,…,Xin, i=1,…,k   are drawn from k(⩾2)k(⩾2) populations Π1,…,ΠkΠ1,…,Πk, respectively, where observations from ΠiΠi have U(0,θi)-distributionU(0,θi)-distribution and let Xi=max(Xi1,…,Xin)Xi=max(Xi1,…,Xin), i=1,…,ki=1,…,k. For selecting the population associated with larger (or smaller) θiθi, i=1,…,k  , we consider the natural selection rule, according to which the population corresponding to the larger (or smaller) XiXi is selected. In this paper, we consider the problem of estimating the parameter θMθM (or θJθJ) of the selected population under the entropy loss function. For k⩾2k⩾2, we generalize the (U,V) methods of Robbins (1988) for entropy loss function and derive the uniformly minimum risk unbiased (UMRU) estimator of θMθM and θJθJ. For k  =2, we obtain the class of all linear admissible estimators of the forms cX(2)cX(2) and cX(1)cX(1) for θMθM and θJθJ, respectively, where X(1)=min(X1,X2)X(1)=min(X1,X2) and X(2)=max(X1,X2)X(2)=max(X1,X2). Also, in estimation of θMθM, we show that the generalized Bayes estimator is minimax and the UMRU estimator is inadmissible. Finally, we compare numerically the risks of the obtained estimators.