The exact distribution of the maximum, minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies

Authors:
Charles J. Corrado
Affiliations:
Faculty of Business & Law, Deakin University, Melbourne, Australia
Venue:
Statistics and Computing
Year:
2011

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Efficient time/space algorithm to compute rectangular probabilities of multinomial, multivariate hypergeometric and multivariate Pólya distributions

Statistics and Computing

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Abstract

The exact distribution of the maximum and minimum frequencies of Multinomial/Dirichlet and Multivariate Hypergeometric distributions 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 range is also obtained. These algorithms satisfy a long-standing need for routines to compute exact Multinomial/Dirichlet and Multivariate Hypergeometric maximum, minimum, and range probabilities in statistical computation libraries and software packages.