Multivariate Liouville distributions

Rameshwar D. Gupta, Donald St P. Richards

Research output: Contribution to journalArticlepeer-review

71 Scopus citations


A random vector (X1, ..., Xn), with positive components, has a Liouville distribution if its joint probability density function is of the formf(x1 + ... + xn)x1a1.1 ... xnan.1 with theai 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.

Original languageEnglish (US)
Pages (from-to)233-256
Number of pages24
JournalJournal of Multivariate Analysis
Issue number2
StatePublished - Dec 1987

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty


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