An alternative proof for a version of cochran's theorem under elliptical settings

Abstract

For an elliptically contoured n × p random matrix Y with mean μ and covariance proportional to ∑_Y, the necessary and sufficient conditions, under which (Y−μ)′W(Y−μ) with nonnegative definite W is generalized Wishart distributed, are obtained by using the higher moments of Y. This version of Cochran's theorem is general as the assumptions on ∑_Y=A⊗∑ with nonnegative definite A and ∑, P(Y−μ)=0, and P(Y≠μ)<1 have been relaxed. An example on two way balanced mixed models is given for illustration ot our main results.