Page 261 - Elements of Distribution Theory

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8.4 Quadratic Forms 247

for all |t| < , for some > 0. It is straightforward to show that the double series converges
absolutely and, hence, the order of summation can be changed:

d d ∞ ∞ d

j
j j j j
log(1 − 2tλ k ) =− (2λ k ) t /j =− λ 2 t /j
k
k=1 k=1 j=1 j=1 k=1
for all |t| < . This implies that
d

j
λ = r, j = 1, 2,..., (8.3)
k
k=1
and, hence, that each λ k is either 0 or 1.
To prove this last fact, consider a random variable W taking value λ k , k = 1, 2,..., d
with probability 1/d; note that, by (8.3), all moments of W are equal to r/d. Since W is
bounded, its moment-generating function exists; by Theorem 4.8 it is given by

r
t j r
∞
1 + = 1 + [exp{t}− 1] = (1 − r/d) + (r/d)exp{t},
d j! d
j=1
which is the moment-generating function of a random variable taking the values 0 and 1
with probabilities 1 − r/d and r/d, respectively. The result follows.

Example 8.10 (Sum of squared independent normal random variables). Let X 1 ,
X 2 ,..., X n denote independent, real-valued random variables, each with a normal distribu-
2
2
tion with mean 0. Let σ = Var(X j ), j = 1,..., n, and suppose that σ > 0, j = 1,..., n.
j
j
Consider a quadratic form of the form
n

2
Q = a j X
j
j=1
where a 1 , a 2 ,..., a n are given constants.
Let X = (X 1 ,..., X n ). Then X has a multivariate normal distribution with covariance
2
matrix , where is a diagonal matrix with jth diagonal element given by σ . Let A
j
T
denote the diagonal matrix with jth diagonal element a j . Then Q = X AX.
It follows from Theorem 8.6 that Q has a chi-squared distribution if and only if A is
2
idempotent. Since A is a diagonal matrix with jth diagonal element given by a j σ ,it
j
follows that Q has a chi-squared distribution if and only if, for each j = 1,..., n, either
2
a j = 0or a j = 1/σ .
j
The same basic approach used in part (iii) of Theorem 8.1 can be used to study the joint
distribution of two quadratic forms, or the joint distribution of a quadratic form and a linear
function of a multivariate normal random vector.
Theorem 8.7. Let X denote a d-dimensional random vector with a multivariate normal
distribution with mean 0 and covariance matrix . Let A 1 and A 2 be d × d nonnegative-
T
definite, symmetric matrices and let Q j = X A j X, j = 1, 2.
(i) If A 1 A 2 = 0 then Q 1 and Q 2 are independent.
T
(ii) Let Y = MX where M is an r × d matrix. If A 1 M = 0 then Y and Q 1 are
independent.