On the Existence of Moments of Ratios of Quadratic Forms
研究了随机向量X的二次型之比T = X'AX / X'BX的混合矩存在的条件，给出了简单且普遍适用的条件，适用于椭圆对称分布（包括多元正态和t分布）及线性约束情形。
We obtain simple and generally applicable conditions for the existence of mixed moments E ([ X ′ AX ]″/[ X ′ BX ] U ) of the ratio of quadratic forms T = X ′ AX / X ′ BX where A and B are n × n symmetric matrices and X is a random n -vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints. If u ≤ v , the precise distribution of X , and in particular the existence of moments of X , is virtually irrelevant to the existence of the mixed moments of T ; if u > v , a prerequisite for existence of the ( u , v )th mixed moment is the existence of the 2( u − v )th moment of X When X is not degenerate, the principal further requirement for the existence of the mixed moment is that B has rank exceeding 2 v .