Page 263 - Elements of Distribution Theory

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8.4 Quadratic Forms 249

and
T T
1 v v .
A 2 = γ 1 v 1 v +· · · + γ r 2 r 2 r 2
Note that

A 1 + A 2 = PDP T
and P is a
where D is a diagonal matrix with diagonal elements λ 1 ,...,λ r 1 ,γ 1 ,...,γ r 2
; recall that r 1 + r 2 = d. Since
matrix with columns e 1 ,..., e r 1 , v 1 ,..., v r 2
T
A 1 + A 2 = PDP = I d ,
and D has determinant

γ = 0,
λ 1 ··· λ r 1 1 ··· γ r 2
it follows that |P| = 0.
Since A 1 e j = λ j e j and (A 1 + A 2 )e j = e j ,it follows that

A 2 e j = (1 − λ j )e j , j = 1,...,r 1 .
That is, either λ j = 1or e j is an eigenvector of A 2 .However,if e j is an eigenvector of
A 2 , then two columns of P are identical, so that |P|= 0; hence, λ j = 1, j = 1,...,r 1 ;
similarly, γ j = 1, j = 1,...,r 2 . Furthermore, all the eigenvectors of A 1 are orthogonal to
the eigenvectors of A 2 and, hence,
A 1 A 2 = 0 and A 2 A 1 = 0.

Also, since (A 1 + A 2 )A 1 = A 1 ,it follows that A 1 is idempotent; similarly, A 2 is idempotent.
It now follows from Theorem 8.6 that Q 1 and Q 2 have chi-squared distributions. To
prove independence of Q 1 and Q 2 , note that
T T
1
Y Y r 1
Q 1 = λ 1 Y Y 1 + ··· + λ r 1 r 1
T
where Y j = e X, j = 1,...,r 1 . Similarly,
j
T T
1 r 2
Q 2 = γ 1 Z Z 1 +· · · + γ r 2 Z Z r 2
T
where Z j = v X. Since each e j is orthogonal to each v j ,it follows from Theorem 8.1
j
that Y i and Z j are independent, i = 1,...,r 1 , j = 1,...,r 2 . Hence, Q 1 and Q 2 are
independent.
In the following corollary, the result in Theorem 8.8 is extended to the case of several
quadratic forms; the proof is left as an exercise. This result is known as Cochran’s Theorem.

Corollary 8.1. Let X denote a d-dimensional random vector with a multivariate normal dis-
tribution with mean 0 and covariance matrix I d . Let A 1 , A 2 ,..., A m be d × d nonnegative-
T
definite, symmetric matrices and let Q j = X A j X, j = 1,..., m, such that
T
X X = Q 1 + Q 2 +· · · + Q m .
Let r j denote the rank of A j ,j = 1,..., m. Q 1 , Q 2 ,..., Q m are independent chi-squared
random variables, such that the distribution of Q j has r j degrees of freedom, j = 1,..., m,
if and only if r 1 +· · · + r m = d.