## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 3 (1984), 1058-1070.

### A General Theory of Asymptotic Consistency for Subset Selection with Applications

#### Abstract

The problem of selecting a random nonempty subset from $k$ populatinos, characterized by $\theta_1, \cdots, \theta_k$ with possible nuisance parameters $\sigma$, is considered using a decision-theoretic approach. The concept of asymptotic consistency is defined as the property that the risk of a procedure at $(\theta, \sigma)$ tends to the minimum loss at $(\theta, \sigma)$. Necessary and sufficient conditions for both pointwise and uniform (on compact sets) consistency for permutation-invariant procedures are derived with general loss functions. Various loss functions when the goal is to select populations with $\theta_i$ close to $\max \theta_j$ are considered. Applications are made to normal populations. It is shown that Gupta's procedure is the only procedure in Seal's class that can be consistent. Other Bayes and admissible procedures are also considered.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 3 (1984), 1058-1070.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346721

**Digital Object Identifier**

doi:10.1214/aos/1176346721

**Mathematical Reviews number (MathSciNet)**

MR751292

**Zentralblatt MATH identifier**

0557.62022

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F07: Ranking and selection

Secondary: 62C99: None of the above, but in this section

**Keywords**

Subset selection asymptotic theory decision-theory invariance consistency

#### Citation

Bjornstad, Jan F. A General Theory of Asymptotic Consistency for Subset Selection with Applications. Ann. Statist. 12 (1984), no. 3, 1058--1070. doi:10.1214/aos/1176346721. https://projecteuclid.org/euclid.aos/1176346721