## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 1 (1984), 336-350.

### Sequential Selection Procedures--A Decision Theoretic Approach

Shanti S. Gupta and Klaus J. Miescke

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 1 (1984), 336-350.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346411

**Digital Object Identifier**

doi:10.1214/aos/1176346411

**Mathematical Reviews number (MathSciNet)**

MR733518

**Zentralblatt MATH identifier**

0539.62035

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F07: Ranking and selection

Secondary: 62F05: Asymptotic properties of tests 62F15: Bayesian inference 62L99: None of the above, but in this section

**Keywords**

Multiple decision procedures sequential procedures screening procedures selection procedures Bayesian analysis

#### Citation

Gupta, Shanti S.; Miescke, Klaus J. Sequential Selection Procedures--A Decision Theoretic Approach. Ann. Statist. 12 (1984), no. 1, 336--350. doi:10.1214/aos/1176346411. https://projecteuclid.org/euclid.aos/1176346411