On Selecting a Subset Containing the Best Population-A Bayesian Approach

This is the authors' abstract. We don't add key points for this paper.

The problem of selecting a subset of k populations π₁, ⋯, πₖ, which contains the "best" population, is considered. The unknown values θ₁, ⋯, θₖ are the characteristics associated with π₁, ⋯, πₖ and the unknown population associated with θₖ = maxᵢθᵢ is called the "best." It is assumed that, given θ = (θ₁, ⋯, θₖ) the pdf of the independent random variables X₁, ⋯, Xₖ belong to a monotone likelihood ratio family, the prior distribution of θ is exchangeable, and the loss function is a linear combination of two components, namely the subset size $|s|$ and the distance between the "best" and the "best" in the selected subset $s,$ i.e., L(θ, s) = c|s| + θₖ - maxj : πⱼ ∈ s θⱼ. It is shown that the Bayes rule depends on at most $(k - 1)$ computable expressions. Some lower and upper bounds on the differences of Bayes risks are given to help reduce the amount of computation for the Bayes rule. If Xᵢ has a normal distribution with mean θᵢ and known variance σ², then it is shown that (i) for $k = 2,$ the Bayes rule with vague prior knowledge and the classical rule are the same if the probability of correct selection, P^, is chosen as a suitable function of $c,$ and (ii) if c/σ 1/π¹/2, then the Bayes rule selects only one population and if .2821 c/σ < 1/π¹/2, then it selects at most two populations. The tables for implementing the Bayes rule for normal populations are also given.