A stochastic ordering of multiple hypergeometric laws: Peakedness of category counts about half the population category sizes is symmetric unimodal in the sample size

Abstract

Suppose X is a frequency vector that follows a central multiple hypergeometric distribution, such as arises in random sampling of an m-category attribute from a finite population without replacement. We call the event where X satisfies a prespecified set of symmetrical—but otherwise arbitrary—interval constraints in each component a symmetric core event. We show that the probability of any symmetric core event—in other words, the multivariate peakedness in the sense of Birnbaum (1948) and Tong (1988)—is symmetric unimodal as a function of the sample size. Two proofs are given. The shorter one relies on a convolution property of ultra-log-concave sequences, which implies that the sequence of peakedness values is log-concave (even for asymmetric rectangular events). The longer, though more elementary, proof does not rely on notions of log-concavity. To illustrate the use of symmetric core events, we analyze a simple yet interesting wager in a sequential card game. Finally, we indicate that the unimodality result for symmetric core events is pivotal in proving a certain variance reduction inequality involving multinomial frequencies subject to arbitrary interval censoring.