Suppose that the random vector (X1, ..., Xq) follows a Dirichlet distribution on q+ with parameter (p1, ..., pq)[set membership, variant]q+. For f1, ..., fq>0, it is well-known that (f1X1+...+fqXq)-(p1+...+pq)=f-p11...f-pqq. 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-r positive-definite real symmetric matrices. For x[set membership, variant][Omega]r and 1[less-than-or-equals, slant]j[less-than-or-equals, slant]r, let detj x denote the jth principal minor of x. For s=(s1, ..., sr)[set membership, variant]r, the generalized power function of x[set membership, variant][Omega]r is the function [Delta]s(x)=(det1 x)s1-s2 (det2 x)s2-s3...(detr-1 x)sr-1-sr (detr x)sr; further, for any t[set membership, variant], we denote by s+t the vector (s1+t, ..., sr+t). Suppose X1, ..., Xq[set membership, variant][Omega]r are random matrices such that (X1, ..., Xq) follows a multivariate Dirichlet distribution with parameters p1, ..., pq. Then we evaluate the expectation [[Delta]s1(X1)...[Delta]sq(Xq) [Delta]s1+...+sq+p((a+f1X1+...+fqXq)-1)], where a[set membership, variant][Omega]r, p=p1+...+pq, f1, ..., fq>0, and s1, ..., sq each belong to an appropriate subset of r+. The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the Xj's are singular. Our derivation utilizes the framework of symmetric cones, so that our results are valid for multivariate Dirichlet distributions on all symmetric cones.
