Multivariate versions of Cochran theorems Original Research Article

A general easily verifiable Cochran theorem is obtained for a normal random matrix Y with mean μ and covariance ΣY which may be singular and may not be of the form A circle times operator Σ, where Σ is the population covariance: {Y′WiY}i=1L (with nonnegative definite Wiʹs) is an independent family of Wishart WY(mi, Σ, λi) random matrix YiWiY if and only if for W = Σi=1L Wi, (W circle times operator I) Σr (W circle times operator I) is of the form C circle times operator Σ with CW+C = C and for all distinet i, j = 1,2,…, L, mi= r(W+CW+Wi), WiW+ CW+ Wj = 0 and λi = μ′Wiμ = μiWiW+CW+Wiμ.