Multivariate Liouville distributions

Authors:
Rameshwar D. Gupta;Donald St. P. Richards
Affiliations:
University of New Brunswick, St. John, New Brunswick ,CanadaE2L 4L5;University of North CarolinaUSA
Venue:
Journal of Multivariate Analysis
Year:
1987

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Abstract

A random vector (X"1, ..., X"n), with positive components, has a Liouville distribution if its joint probability density function is of the formf(x"1 + ... + x"n)x"1^a^"^1^.^1 ... x"n^a^"^n^.^1 with thea"i all positive. Examples of these are the Dirichlet and inverted Dirichlet distributions. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results pertain to stochastic representations, transformation properties, complete neutrality, marginal and conditional distributions, regression functions, and total positivity and reverse rule properties. Further, these topics are utilized in various characterizations of the Dirichlet and inverted Dirichlet distributions. Matrix analogs of the Liouville distributions are also treated, and many of the results obtained in the vector setting are extended appropriately.