The Exact Distribution of the Maximum, Minimum and the Range of Multinomial/Dirichlet and Multivariate Hypergeometric Frequencies

Abstract

The exact joint distribution of the maximum and minimum of a multinomial distribution of n balls in m urns is compactly represented as a product of stochastic matrices. This representation does not require equal urn probabilities, is invariant to urn order, and permits rapid calculation of exact probabilities. The exact distribution of the multinomial range is also derived. An application to the September effect in stock returns is presented.

Keywords: Dirichlet multinomial, Multinomial maximum minimum range, Multinomial outliers and inliers, September effect, Stochastic matrix

JEL Classification: C10, C12, C14, C63