## The Annals of Statistics

### Is the Selected Population the Best?

#### Abstract

Random variables $X_i \sim N(\theta_i, 1), i = 1,2,\cdots, k$, are observed. Suppose $X_S$ is the largest observation. If the inference $\theta_S > \max_{i\neq S}\theta_i$ is made whenever $X_S - \max_{i\neq S}X_i > c$, then the probability of a false inference is maximized when two $\theta_i$ are equal and the rest are $-\infty$. Equivalently, the inference can be made whenever a two-sample two-sided test for difference of means, based on the largest two observations, would reject the hypothesis of no difference. The result also holds in the case of unknown, estimable, common variance, and in fact for location families with monotone likelihood ratio.

#### Article information

Source
Ann. Statist., Volume 15, Number 1 (1987), 456-461.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176350281

Digital Object Identifier
doi:10.1214/aos/1176350281

Mathematical Reviews number (MathSciNet)
MR885752

Zentralblatt MATH identifier
0623.62021

JSTOR