COCHRAN THEOREMS FOR A MULTIVARIATE ELLIPTICALLY CONTOURED MODEL

Let Y be a multivariate elliptically contoured n x p random matrix with an MEC(nxp)(mu, Sigma(Y), phi) distribution and P(Y = mu) < 1. Let i = 1, 2,..., L, W-i be an n x n nonnegative definite (n.n.d.) matrix, and m(i) epsilon {1, 2, ...}, Q(i)(Y) = (Y - mu)' W-i(Y - mu) and Sigma(not equal 0) be a p x p n.n.d. matrix. Then (a) and (b) are equivalent: (a) (Q(1)(Y), Q(2)(Y),...,Q(L)(Y)) similar to GW(p)(m(1), m(2),...,mL;n - Sigma(i=1)(L)m(i);Sigma;phi) (b) For some n.n.d. n x n matrix A and for any distinct i, j = 1, 2,..., L, (i) (W(i)xI(p)) (Sigma(Y) - Ax Sigma) (W(i)xI(p)) = 0, (ii) AW(i)AW(i) = AW(i), r(AW(i)) = m(i), (iii) W(i)AW(j) = 0 and (iv) (W(i)xI(p))Sigma(Y)(W(j)xI(p)) = 0. Moreover if r(W-i) = m(i) for each i, then (a') and (b') are equivalent: (a') (Q(1)(Y),Q(2)(Y),...,Q(L)(Y))similar to GW(p)(r(W-1),...,r(W-L); n - Sigma(i=1)(L)r(W-i); Sigma; phi). (b') For any distinct i,j = 1, 2,..., L, (i) (W(i)xI(p))Sigma(Y)(W(i)xI(p)) = W(i)x Sigma and (ii) (W(i)xI(p))Sigma(Y)(W(i)xI(p)) = 0. Applications are given for certain MANOVA models and multivariate components of variance models. The above results are general in that Sigma(Y) is no longer required to have the form Ax Sigma.