An expectation formula for the multivariate dirichlet distribution

Suppose that the random vector (X₁, …, X_q) follows a Dirichlet distribution on R^q ₊ with parameter (p₁, …, p_q)∈R^q ₊. For f₁, …, f_q>0, it is well-known that E(f₁X₁+…+f_qX_q) ^-(p1+…+pq)=f^-p1 ₁…f^-pq _q. In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let Ω_r denote the cone of all r×r positive-definite real symmetric matrices. For x∈Ω_r and 1≤j≤r, let det_jx denote the jth principal minor of x. For s=(s₁, …, s_r)∈R^r, the generalized power function of x∈Ω_r is the function Δ_s(x)=(det₁x)^s1-s2(det₂x) ^s2-s3…(det_r-1x)^sr-1-sr(det_rx) ^sr; further, for any t∈R, we denote by s+t the vector (s₁+t, …, s_r+t). Suppose X₁, …, X_q∈Ω_r are random matrices such that (X₁, …, X_q) follows a multivariate Dirichlet distribution with parameters p₁, …, p_q. Then we evaluate the expectation E[Δ_s1(X₁)…Δ_sq(X _q)Δ_s1+…+sq+p((a+f₁X ₁+…+f_qX_q)^-1)], where a∈Ω_r, p=p₁+…+p_q, f₁, …, f_q>0, and s₁, …, 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 X_j'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.