## The Annals of Mathematical Statistics

### An Approximation to the Sample Size in Selection Problems

Edward J. Dudewicz

#### Abstract

Let $f(\mathbf{x} \mid P_1)$ be the $\operatorname{pdf}$ of a $(k - 1)$-dimensional normal distribution with zero means, unit variances, and correlation matrix $P_1$. Consider the integral, for $\delta > 0$, \begin{equation*}\tag{1}\int^\infty_{-\delta} \cdots \int^\infty_{-\delta} f(\mathbf{x} \mid P_1)dx \cdots dx_{k-1} = \alpha(\delta), \text{say}.\end{equation*} Assume that no element of $P_1$ is a function of $\delta$. Note that $\alpha(\delta)$ is an increasing function of $\delta$ and $\alpha(\delta) \rightarrow 1$ as $\delta \rightarrow \infty$. The problem is to obtain an approximation to $\delta$, for a large specified value, $\alpha$, of $\alpha(\delta)$. This is given by the theorem of Section 1. This result is used to obtain approximations to the sample size in a selection procedure of Bechhofer and in a problem of selection from a multivariate normal population. The closeness of the approximation is illustrated for the procedure of Bechhofer (Table 1).

#### Article information

Source
Ann. Math. Statist. Volume 40, Number 2 (1969), 492-497.

Dates
First available in Project Euclid: 27 April 2007

http://projecteuclid.org/euclid.aoms/1177697715

Digital Object Identifier
doi:10.1214/aoms/1177697715

Mathematical Reviews number (MathSciNet)
MR246449

Zentralblatt MATH identifier
0177.22506

JSTOR