## The Annals of Statistics

### 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

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