An Approximation to the Sample Size in Selection Problems

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).