Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

560 Inferences About Normal Linear Models

Example 9.8.3.Based on this last theorem, we can obtain quickly the distri-
bution of the sample variance when sampling from a normal distribution. Sup-
poseX 1 ,X 2 ,...,Xnare iidN(μ, σ^2 ). LetX=(X 1 ,X 2 ,...,Xn)′.ThenXhas a
Nn(μ 1 ,σ^2 I) distribution, where 1 denotes an×1 vector with all components equal
to 1. LetS^2 =(n−1)−^1

∑n

i=1(Xi−X)

(^2). Then by Example 9.8.1, we can write
(n−1)S^2
σ^2
=σ−^2 X′
(
I−
1
n
J
)
X=σ−^2 (X−μ 1 )′
(
I−
1
n
J
)
(X−μ 1 ),
where the last equality holds because
(
I−n^1 J
)
1 = 0. Because the matrixI−^1 nJis
idempotent, tr (I−n^1 J)=n−1, andX−μ 1 isNn( 0 ,σ^2 I), it follows from Theorem
9.8.4 that (n−1)S^2 /σ^2 has aχ^2 (n−1) distribution.
Remark 9.8.3.If the normal distribution in Theorem 9.8.4 isNn(μ,σ^2 I), the
conditionA^2 =Aremains a necessary and sufficient condition thatQ/σ^2 have a
chi-square distribution. In general, however,Q/σ^2 is not centralχ^2 (r) but instead,
Q/σ^2 has a noncentral chi-square distribution ifA^2 =A. The number of degrees
of freedom isr, the rank ofA, and the noncentrality parameter isμ′Aμ/σ^2 .If
μ=μ 1 ,thenμ′Aμ=μ^2
∑
i,jaij,whereA=[aij]. Then, ifμ^ = 0, the condi-
tionsA^2 =Aand
∑
i,j
aij= 0 are necessary and sufficient conditions thatQ/σ^2
be centralχ^2 (r). Moreover, the theorem may be extended to a quadratic form in
random variables which have a multivariate normal distribution with positive def-
inite covariance matrixΣ; here the necessary and sufficient condition thatQhave
a chi-square distribution isAΣA=A. See Exercise 9.8.9.
EXERCISES
9.8.1.LetQ=X 1 X 2 −X 3 X 4 ,whereX 1 ,X 2 ,X 3 ,X 4 is a random sample of size 4
from a distribution that isN(0,σ^2 ). Show thatQ/σ^2 does not have a chi-square
distribution. Find the mgf ofQ/σ^2.
9.8.2.LetX′=[X 1 ,X 2 ] be bivariate normal with matrix of meansμ′=[μ 1 ,μ 2 ]
and positive definite covariance matrixΣ.Let
Q 1 =
X 12
σ 12 (1−ρ^2 )
− 2 ρ
X 1 X 2
σ 1 σ 2 (1−ρ^2 )

• X 22
  σ^22 (1−ρ^2 )
  .
  Show thatQ 1 isχ^2 (r, θ) and findrandθ. When and only when doesQ 1 have a
  central chi-square distribution?
  9.8.3.LetX′=[X 1 ,X 2 ,X 3 ] denote a random sample of size 3 from a distribution
  that isN(4,8) and let
  A=
  ⎛
  ⎝
  1
  2 0
  1
  2
  010
  1
  2 0
  1
  2
  ⎞
  ⎠.
  LetQ=X′AX/σ^2.