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

Permanent link to this document
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
links.jstor.org

Citation

Dudewicz, Edward J. An Approximation to the Sample Size in Selection Problems. Ann. Math. Statist. 40 (1969), no. 2, 492--497. doi:10.1214/aoms/1177697715. http://projecteuclid.org/euclid.aoms/1177697715.