An expectation formula for the multivariate dirichlet distribution

JOURNAL OF MULTIVARIATE ANALYSIS

卷:77

Article

Letac, GR ; Massam, H ; Richards, D

关键词: Dirichiet distribution;    gamma distribution;    Gaussian hypergeometric function;    generalized power function;    Jordan algebra;    Laplace transform;    Lauricella function;    multivariate beta distribution;    multivariate gamma function;    Riesz measure;    symmetric cone;    Wishart distribution;   

DOI  :  10.1006/jmva.2000.1928

来源: Elsevier

【 摘 要 】

Suppose that the random vector (X-1,....X-q) follows a Dirichlet distribution on R-+(q) with parameter (p(1),....p(q))is an element of R-+(q). For f(1),...,f(q) > 0, it is well-known that E (f(1) X-1 +...+ f(q)X(q))(-(p1+...+pq)) = f(1)(-pq)...f(q)(-pq). In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let Omega (r) denote the cone of all r x r positive-definite real symmetric matrices. For x is an element of Omega (r) and 1 less than or equal to j less than or equal to r, let det (j)x denote the jth principal minor of x. For s = (s(1),...,s(r)) is an element of R-r, the generalized power function of x is an element of Omega (r) is the function Delta (s)(x)0 = (det(1) x)(s1-s2) (det(2) x)(s2-s3) ...(()det(r-1)x)(sr-1-sr)(det(r) x)(sr); further, for any t is an element of R. We denote by s + t the vector (s(1) + t,...,s(r)+t). Suppose X-1,...X-q is an element of Omega (r), are random matrices such that (X-1,....,X-q) follows a multivariate Dirichlet distribution With parameters p(1),...,p(q). Then we evaluate the expectation E [Delta (s1)(X-1)...Delta (s)(X-q) Delta (s1) (+) .. (+sq + p) ((a + f(1)X(1) + ... + f(q)X(q))(-1))], where a is an element of Omega (r), p = p(1) + ... +p(q), f(1),...f(q) > 0, and s(1),...,s(q) each belong to an appropriate subset of R-+(r). The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the Xfs are singular. Our derivation utilizes the framework of symmetric cones. so that our results are valid for multivariate Dirichlet distributions on all symmetric cones. (C) 2001 Academic Press.

【 授权许可】

Free