Page 259 - Elements of Distribution Theory

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

8.4 Quadratic Forms 245

Thus, the previous example shows that the sample variance is a quadratic form. Similarly,
the sums of squares arising in regression analysis and the analysis of variance can also
generally be expressed as quadratic forms. Quadratic forms also arise in the approximation
of the distribution of certain test statistics.
T
Since the term in the exponent of the multivariate normal distribution, −x −1 x/2, is
a quadratic form in x,itisa relatively simple matter to determine the moment-generating
function and, hence, the cumulant-generating function, of a quadratic form.

Theorem 8.5. Let X denote a d-dimensional random vector with a multivariate nor-
mal distribution with mean vector 0 and covariance matrix , | | > 0. Let A be a
T
d × d nonnegative-definite, symmetric matrix and let Q = X AX. Then Q has cumulant-
generating function
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. The jth cumulant of Q is given by
d

j−1 j
2 ( j − 1)!λ .
k
k=1
T
Proof. Consider E[exp{tX AX}], which is given by

d 1 1 T −1 T
(2π) − 2 | | − 2 exp − x x exp{tx Ax} dx
R d 2

d 1 1 T −1
= (2π) − 2 | | − 2 exp − x [ − 2tA]x dx.
R d 2
Note that | −1 − 2tA| is a continuous function of t and is positive for t = 0; hence, there
exists a δ> 0 such that
| −1 − 2tA| > 0 for |t| <δ.

Thus, for |t| <δ,

d 1 1 T −1 1 −1 1
(2π) − 2 | | − 2 exp − x [ − 2tA]x dx =| | − 2 | − 2tA| − 2
R d 2
1 1 1
=|I d − 2t 2 A 2 | − 2 .
1 1
Let B = 2 A 2 . Note that B is a symmetric nonnegative-definite matrix so that we may
write
B = PDP T

where P is an orthogonal matrix and D is a diagonal matrix with diagonal elements given
by the eigenvalues of B or, equivalently, the eigenvalues of A.It follows that

n

T T
|I d − 2tP DP |=|P(I d − 2tD)P |=|I d − 2tD|= (1 − 2tλ j )
j=1