On a Statistic which Arises in Selection and Ranking Problems

Abstract

The statistic $y = (x_{\lbrack p\rbrack} - x)/s_\nu$ is studied where $x_{\lbrack p\rbrack}$ is the maximum of $p$ normal independent chance variables with common mean and common unknown variance $\sigma^2, x$ is another independent normal chance variable with the same mean and the same variance $\sigma^2,$ and $s^2_\nu$ (distributed as $\sigma^2\chi^2_\nu/\nu$ with $\nu$ degrees of freedom) is an estimate of the common variance which is independent of each one of the above $p + 1$ chance variables. Several different methods are proposed and studied for computing the probability integral of $y$ and percentage points of $y$; in addition, a method for computing percentage points without first computing the probability integral of $y$ is considered. A table of (upper) percentage points of $y$ is given as Table I at the end of the paper. Applications of the statistic $y$ to several ranking and selection problems are mentioned in Section 2. Moments of $y$ are given in Section 3. In Section 7 it is shown that Table I can be used to obtain an approximation and bounds to the percentage points of a related statistic.