## Abstract

Suppose that the random vector (X_{1}, …, X_{q}) follows a Dirichlet distribution on R^{q} _{+} with parameter (p_{1}, …, p_{q})∈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^{-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_{j}x denote the jth principal minor of x. For s=(s_{1}, …, s_{r})∈R^{r}, the generalized power function of x∈Ω_{r} is the function Δ_{s}(x)=(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∈R, we denote by s+t the vector (s_{1}+t, …, s_{r}+t). Suppose X_{1}, …, X_{q}∈Ω_{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[Δ_{s1}(X_{1})…Δ_{sq}(X _{q})Δ_{s1+…+sq+p}((a+f_{1}X _{1}+…+f_{q}X_{q})^{-1})], where a∈Ω_{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 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.

Original language | English (US) |
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Pages (from-to) | 117-137 |

Number of pages | 21 |

Journal | Journal of Multivariate Analysis |

Volume | 77 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2001 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty