Maximizing the Probability That Adjacent Order Statistics of Samples from Several Populations Form Overlapping Intervals

Abstract

Let samples of size $n$ be drawn from each of $k$ univariate continuous cumulative distribution functions on the same real line, and consider the intersection of the $k$ intervals between the $r$th and $(r + 1)$st order statistics in the several samples. Then, to maximize the probability that that intersection be nonempty the distributions should be identical. Furthermore, for each sample, consider two intervals--that between the $r$th and $(r + 1)$st and that between the $s$th and $(s + 1)$st order statistics-then to maximize the probability that both the intersection of the "$r$" intervals and the intersection of the "$s$" intervals be nonempty, the distributions again should be identical and the value of the maximum probability is $\frac{\binom{n}{r}^k \binom{n - r}{s - r}^k}{\binom{kn}{kr}\binom{k\lbrack n - r\rbrack}{k\lbrack s - r\rbrack}}, \qquad r \leqq s.$ Some possible directions for generalization are discussed. The problem arose in connection with a sociological study of interaction behavior in small groups. The results make it possible to provide a test of the hypothesis that several samples of the same size are randomly drawn from possibly different populations, against the alternative that the samples are not independently and randomly drawn from distributions. For example, suppose we observe the frequency of a particular sort of interaction for each member of five groups of size six. Suppose the five men with the highest frequencies each belong to different group. Then we can say (ignoring discreteness) that an event has occurred whose probability under random sampling is at most 144/2639 or about 0.055. (The statistic would have but two values, either the five highest belong to different groups, or they do not. Such a test would be especially appropriate if group structure were thought to develop automatically certain specialized functions in members.) However, the main interest in this paper is in the problem in probability.