Problem of Finding the Largest Normal Mean under Heteroscedasticity

摘要

Let P sub 1,..., P sub k be k approximately equal to 3 given normal populations with unknown means theta sub 1,..., theta sub k, and a common known variance sigma squared. Let X sub 1,..., X sub k be the sample means of k independent samples o sizes n sub 1,...,n sub k from these populations. To find the population with the largest mean, one usually applies the natural rule d sub N, which selects in terms of the largest sample mean. In this paper, the performance of this rule is studied under 0 - 1 loss. It is shown that d sub n is minimax if and only if n sub 1 = ...= n sub k. d sub N is seen to perform weakly whenever the parameters theta sub 1,..., theta sub k are close together. Several alternative selection rules are derived in a Bayesian approach which seem to be reasonable competitors to d sub N, worth comparing with d sub N in a future simulation study. (Author)