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

• Published in: Journal of statistical planning and inference 2012-07, Vol.142 (7), p.2190-2202
• Main Authors: Nematollahi, N., Motamed-Shariati, F.
• Format: Article
• Language: English
• Subjects: Admissible estimators; Entropy loss function; Estimation after selection; Minimax estimator; UMRU estimator; Uniform distribution
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2012.01.016

Suppose independent random samples Xi1,…,Xin, i=1,…,k are drawn from k(⩾2) populations Π1,…,Πk, respectively, where observations from Πi have U(0,θi)-distribution and let Xi=max(Xi1,…,Xin), i=1,…,k. For selecting the population associated with larger (or smaller) θi, i=1,…,k, we consider the natural selection rule, according to which the population corresponding to the larger (or smaller) Xi is selected. In this paper, we consider the problem of estimating the parameter θM (or θJ) 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 θM and θJ. For k=2, we obtain the class of all linear admissible estimators of the forms cX(2) and cX(1) for θM and θJ, respectively, where X(1)=min(X1,X2) and X(2)=max(X1,X2). Also, in estimation of θM, 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.