Moment properties of the multivariate Dirichlet distributions

Rameshwar D. Gupta, Donald Richards

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let X1, ⋯, Xn be real, symmetric, m × m random matrices; denote by Im the m × m identity matrix; and let a1, ⋯, an be fixed real numbers such that aj > (m-1)/2, j=1, ⋯, n. Motivated by the results of J. G. Mauldon (Ann. Math. Statist. 30 (1959), 509-520) for the classical Dirichlet distributions, we consider the problem of characterizing the joint distribution of (X1, ⋯, Xn) subject to the condition that E Im - ∑j=1n TjXj−(a1+⋯+an) = ∏j=1n Im - Tj−aj for all m × m symmetric matrices T1, ⋯, Tn in a neighborhood of the m × m zero matrix. Assuming that the joint distribution of (X1, ⋯, Xn) is orthogonally invariant, we deduce the following results: each Xj is positive-definite, almost surely; X1 +⋯ + Xn = Im, almost surely; the marginal distribution of the sum of any proper subset of X1, ⋯, Xn is a multivariate beta distribution; and the joint distribution of the determinants (X1, ⋯, Xn) is the same as the joint distribution of the determinants of a set of matrices having a multivariate Dirichlet distribution with parameter (a1, ⋯, an). In particular, for n = 2 we obtain a new characterization of the multivariate beta distribution.

Original languageEnglish (US)
Pages (from-to)240-262
Number of pages23
JournalJournal of Multivariate Analysis
Volume82
Issue number1
DOIs
StatePublished - Jan 1 2002

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Dirichlet Distribution
Multivariate Distribution
Joint Distribution
Moment
Beta distribution
Determinant
Zero matrix
Proper subset
Unit matrix
Marginal Distribution
Random Matrices
Symmetric matrix
Positive definite
Deduce
Denote
Invariant
Dirichlet
Joint distribution

All Science Journal Classification (ASJC) codes

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

Cite this

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title = "Moment properties of the multivariate Dirichlet distributions",
abstract = "Let X1, ⋯, Xn be real, symmetric, m × m random matrices; denote by Im the m × m identity matrix; and let a1, ⋯, an be fixed real numbers such that aj > (m-1)/2, j=1, ⋯, n. Motivated by the results of J. G. Mauldon (Ann. Math. Statist. 30 (1959), 509-520) for the classical Dirichlet distributions, we consider the problem of characterizing the joint distribution of (X1, ⋯, Xn) subject to the condition that E Im - ∑j=1n TjXj−(a1+⋯+an) = ∏j=1n Im - Tj−aj for all m × m symmetric matrices T1, ⋯, Tn in a neighborhood of the m × m zero matrix. Assuming that the joint distribution of (X1, ⋯, Xn) is orthogonally invariant, we deduce the following results: each Xj is positive-definite, almost surely; X1 +⋯ + Xn = Im, almost surely; the marginal distribution of the sum of any proper subset of X1, ⋯, Xn is a multivariate beta distribution; and the joint distribution of the determinants (X1, ⋯, Xn) is the same as the joint distribution of the determinants of a set of matrices having a multivariate Dirichlet distribution with parameter (a1, ⋯, an). In particular, for n = 2 we obtain a new characterization of the multivariate beta distribution.",
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Moment properties of the multivariate Dirichlet distributions. / Gupta, Rameshwar D.; Richards, Donald.

In: Journal of Multivariate Analysis, Vol. 82, No. 1, 01.01.2002, p. 240-262.

Research output: Contribution to journalArticle

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