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

Abstract Suppose independent random samples X i 1 , … , X in , i=1,…,k are drawn from k ( ⩾ 2 ) populations Π 1 , … , Π k , respectively, where observations from Π i have U ( 0 , θ i ) - distribution and let X i = max ( X i 1 , … , X in ) , i = 1 , … , k . For selecting the population associated with larger (or smaller) θ 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 θ M (or θ 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 θ M and θ J . For k=2, we obtain the class of all linear admissible estimators of the forms cX ( 2 ) and cX ( 1 ) for θ M and θ J , respectively, where X ( 1 ) = min ( X 1 , X 2 ) and X ( 2 ) = max ( X 1 , X 2 ) . Also, in estimation of θ 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.