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

• Published in: Sequential analysis Vol. 43; no. 4; pp. 417 - 431
• Main Author: Levin, Bruce
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01.10.2024; Taylor & Francis Ltd
• Subjects: Card games; Concavity; Interval censoring; log-concavity; multinomial distribution; multiple hypergeometric distribution; multivariate peakedness; Random sampling; Sequences; Stochastic models; stochastic ordering; symmetric core events; ultra-log-concavity; variance reduction due to interval censoring
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2024.2379901

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.