## The Annals of Statistics

- Ann. Statist.
- Volume 9, Number 4 (1981), 777-791.

### A Class of Schur Procedures and Minimax Theory for Subset Selection

#### Abstract

The problem of selecting a random subset of good populations out of $k$ populations is considered. The populations $\Pi_1, \cdots, \Pi_k$ are characterized by the location parameters $\theta_1, \cdots, \theta_k$ and $\Pi_i$ is said to be a good population if $\theta_i > \max_{1 \leq j\leq k}\theta_j - \Delta$, and a bad population if $\theta_i \leq \max_{1 \leq j \leq k} \theta_j - \Delta$, where $\Delta$ is a specified positive constant. A theory for a special class of procedures, called Schur procedures, is developed, and applied to certain minimax problems. Subject to controlling the minimum expected number of good populations selected or the probability that the best population is in the selected subset, procedures are derived which minimize the expected number of bad populations selected or some similar criterion. For normal populations it is known that the classical "maximum-type" procedures has certain minimax properties. In this paper, two other procedures are shown to have several minimax properties. One is the "average-type" procedure. The other procedure has not previously been considered as a serious contender.

#### Article information

**Source**

Ann. Statist., Volume 9, Number 4 (1981), 777-791.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345518

**Digital Object Identifier**

doi:10.1214/aos/1176345518

**Mathematical Reviews number (MathSciNet)**

MR619281

**Zentralblatt MATH identifier**

0475.62016

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F07: Ranking and selection

Secondary: 62C99: None of the above, but in this section 26A51: Convexity, generalizations

**Keywords**

Subset selection location model Schur-concave functions minimax procedures

#### Citation

Bjornstad, Jan F. A Class of Schur Procedures and Minimax Theory for Subset Selection. Ann. Statist. 9 (1981), no. 4, 777--791. doi:10.1214/aos/1176345518. https://projecteuclid.org/euclid.aos/1176345518