Distributions of Quadratic Forms of Multivariate Normal Random Variables
As we will see in Section 5.6, quadratic forms of multivariate normal random variables play a key role in statistical testing theory. The two most important results are stated in Theorems 5.9 and 5.10:
Theorem 5.9: Let X be distributed Nn(0, Y), where Y is nonsingular. Then XT£-1 X is distributed as x„.
Proof: Denote Y = (Y1,…, Yn)T = Y- /2X. Then Yis n-variate, standard normally distributed; hence, Y1,…,Yn are independent identically distributed (i. i.d.) N(0, 1), and thus, XT Y-1 X = YTY = Ynj=1 Y2 – x2. Q. E.D.
The next theorem employs the concept of an idempotent matrix. Recall from Appendix I that a square matrix M is idempotent if M2 = M. If M is also symmetric, we can write M = QA QT, where A is the diagonal matrix of eigenvalues of M and Q is the corresponding orthogonal matrix of eigenvectors. Then M2 = M implies A2 = A; hence, the eigenvalues of M are either 1 or 0. If all eigenvalues are 1, then A = I; hence, M = I. Thus, the only nonsingular symmetric idempotent matrix is the unit matrix. Consequently, the concept of a symmetric idempotent matrix is only meaningful if the matrix involved is singular.
The rank of a symmetric idempotent matrix M equals the number of nonzero eigenvalues; hence, trace(M) = trace( Q A QT) = trace(A QT Q) = trace(A) = rank(A) = rank(M), where trace(M) is defined as the sum of the diagonal elements of M. Note that we have used the property trace(AB) = trace(BA) for conformable matrices A and B.
Theorem 5.10: LetXbe distributed Nn (0, I), and let M be a symmetric idem – potent n x n matrix of constants with rank k. Then XJMX is distributed xf.
Proof: We can write
where Q is the orthogonal matrix of eigenvectors. Because Y = (Y1Yn )T = QTX ~ Nn(0, I), we now have