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

Suppose independent random samples X-i1, ... X-in i = 1, ..., k are drawn from k(>= 2) populations Pi(1), ... ,Pi(k), respectively, where observations from Pi(i) have U(0, theta(i))-distribution and let X-i = max(X-i1, ... ,X-in), i = 1, ... ,k For selecting the population associated with larger (or smaller) theta(i), i = 1, ... ,k we consider the natural selection rule, according to which the population corresponding to the larger (or smaller) X-i is selected. In this paper, we consider the problem of estimating the parameter theta(M) (or (theta(J)) of the selected population under the entropy loss function. For k >= 2, we generalize the (U,V) methods of Robbins (1988) for entropy loss function and derive the uniformly minimum risk unbiased (UMRU) estimator of theta(M) and theta(J). For k = 2, we obtain the class of all linear admissible estimators of the forms cX((2)) and cX((1)) for theta(M) and theta(J), respectively, where X-(1) = min(X-1, X-2) and X-(2) = max(X-1, X-2). Also, in estimation of theta(M), 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. (C) 2012 Elsevier B.V. All rights reserved.