Page 260 - Elements of Distribution Theory

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052184472Xc08 CUNY148/Severini May 24, 2005 17:54

246 Normal Distribution Theory

where λ 1 ,...,λ d denote the eigenvalues of B. Hence, Q has cumulant-generating function
d
1
K Q (t) =− log(1 − 2tλ k ), |t| <δ;
2
k=1
the expression for the cumulants follows easily from differentiating K Q .

Recall that the sum of squared independent standard normal random variables has a
chi-squared distribution (Example 3.7). That is, if Z is a d-dimensional random vector with
a multivariate normal distribution with mean 0 and covariance matrix given by I d , then
T
T
the quadratic form Z Z has a chi-squared distribution. A general quadratic form X AX
has a chi-squared distribution if it can be rewritten as the sum of squared independent
standard normal random variables. A necessary and sufficient condition for this is given in
the following theorem.

Theorem 8.6. Let X denote a d-dimensional random vector with a multivariate normal
distribution with mean 0 and covariance matrix . Let A be a d × d nonnegative-definite,
T
symmetric matrix and let Q = X AX.Q has a chi-squared distribution if and only if A
is idempotent. The degrees of freedom of the chi-squared distribution is the trace of A.

Proof. From Theorem 8.5, the cumulant-generating function of Q is
d
1
K Q (t) =− log(1 − 2tλ k ), |t| <δ,
2
k=1
where λ 1 ,...,λ d are the eigenvalues of A and δ> 0. If A is idempotent, then each λ k
is either 0 or 1. Suppose that r eigenvalues are 1. Then
1
K Q (t) =− r log(1 − 2t), |t| <δ,
2
which is the cumulant-generating function of a chi-squared random variable with r degrees
of freedom. Hence, by Theorem 4.9, Q has a chi-squared distribution with r degrees of
freedom.
Now suppose that
1
K Q (t) =− r log(1 − 2t)
2
for some r; that is, suppose that
d

log(1 − 2tλ k ) = r log(1 − 2t), |t| <δ,
k=1
for some δ> 0. For any positive number λ,
∞

j j
log(1 − 2tλ) =− (2λ) t /j,
j=1
for sufficiently small |t|. Hence,

d d ∞

j j
log(1 − 2tλ k ) =− (2λ k ) t /j
k=1 k=1 j=1