Sequential Selection Procedures--A Decision Theoretic Approach

Abstract

Let $\pi_1,\cdots, \pi_k$ be given populations which are associated with unknown real parameters $\theta_1,\cdots, \theta_k$ from a common underlying exponential family $\mathscr{F}$. Permutation invariant sequential selection procedures are considered to find good populations (i.e. those which have large parameters), where inferior populations are intended to be screened out at the earlier stages. The natural terminal decisions, i.e. decisions which are made in terms of largest sufficient statistics, are shown to be optimum in terms of the risk, uniformly in $(\theta_1,\cdots, \theta_k)$, under fairly general loss assumptions. Similar results with respect to subset selections within stages are established under the additional assumption that $\mathscr{F}$ is strongly unimodal (i.e. $\log$-concave). The results are derived in the Bayes approach under symmetric priors. Backward induction as well as the concept of decrease in transposition (DT) by Hollander, Proschan and Sethuraman (1977) are the main tools which are used in the proofs.