Some empirical Bayes rules for selecting the best population with multiple criteria

Consider k (k⩾2) normal populations whose mean θi and variance σi2 are all unknown. Let ηi be some function of θi and σi2 and ηi is the parameter of main interest. For given control values η0 and σ02, we want to select some population whose associated value of ηi the largest and also it is larger than η0 and whose associated variance is less than or equal to σ02. An empirical Bayes selection rule is proposed which has been shown to be asymptotically optimal with convergence rate of order O((lnN)2/N), where N is the minimum number of past observations at hand in each population. A simulation study is also carried out for the performance of the proposed empirical Bayes selection rule, and it is found satisfactory.