EMPIRICAL BAYES PROCEDURES FOR SELECTING THE BEST POPULATION WITH MULTIPLE CRITERIA

WEN-TAO HUANG^{1} AND YAO-TSUNG LAI^{2}

1Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, R.O.C. 2Graduate Institute of Mathematics, Tamkang University, Tamsui, Taiwan, R.O.C.

(Received December 26, 1996; revised February 4, 1998)

Abstract. Consider k (k ≥ 2) populations whose mean θ_{i} and variance σ_{i}^{2} are all unknown. For given control values θ_{0} and σ_{0}^{2}, we are interested in selecting some population whose mean is the largest in the qualified subset in which each mean is larger than or equal to θ_{0} and whose variance is less than or equal to σ_{0}^{2}. In this paper we focus on the normal populations in details. However, the analogous method can be applied for the cases other than normal in some situations. A Bayes approach is set up and an empirical Bayes procedure is proposed which has been shown to be asymptotically optimal with convergence rate of order O(ln^{2} n/n). A simulation study is carried out for the performance of the proposed procedure and it is found satisfactory.

Key words and phrases: Best population, multiple criteria, asymptotical optimality, empirical Bayes rule, convergence rate.