On selecting the largest success probability under unequal sample sizes (Q1116578)

Let \(\pi_ 1,...,\pi_ k\) be \(k\geq 3\) independent binomial populations, from which \(X_ i\sim B(n_ i,p_ i)\), \(i=1,...,k\), respectively, have been observed. The problem under concern is to find that population which is associated with the largest of the unknown `success probabilities' \(p_ 1,...,p_ k\). Under the `0-1' loss, some linear loss which occurs in gambling, and a general monotone, permutation invariant loss, interesting properties of Bayes rules are studied for priors which are permutation invariant, as well as for priors which are not invariant but have a (DT)-posterior density with respect to some symmetric measure. Examples of independent beta-priors are included.