Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

558 Inferences About Normal Linear Models

BecauseW 1 ,...,Wrare independentN(0,1) random variables,W 12 ,...,Wr^2 are
independentχ^2 (1) random variables. Thus the mgf ofQis

E[exp{tQ}]=E

[

exp

{r

∑

i=1

tλiWi^2

}]

=

∏r

i=1

E[exp{tλiWi^2 }]=

∏r

i=1

(1− 2 tλi)−^1 /^2. (9.8.13)

The last equality holds if we assume that|t|< 1 /(2λ∗), whereλ∗=max 1 ≤i≤r|λi|;
see Exercise 9.8.6. To obtain the second form in (9.8.10), recall that the determinant
of an orthogonal matrix is 1. The result then follows from

|I− 2 tA|=|Γ′Γ− 2 tΓ′ΛΓ| = |Γ′(I− 2 tΛ)Γ|

= |I− 2 tΛ|=

{r

∏

i=1

(1− 2 tλi)−^1 /^2

}− 2

.

Example 9.8.2.To illustrate this theorem, supposeXi,i=1, 2 ,...,n,arein-
dependent random variables withXidistributed asN(μi,σ^2 i),i=1, 2 ,...,n,re-
spectively. LetZi=(Xi−μi)/σi. We know that

∑n

i=1Z

2

i has aχ

(^2) distribution
withndegrees of freedom. To illustrate Theorem 9.8.2, letZ′=(Z 1 ,...,Zn). Let
Q=Z′IZ. Hence the symmetric matrix associated withQis the identity matrixI,
which hasneigenvalues, all of value 1; i.e.,λi≡1. By Theorem 9.8.2, the mgf of
Qis (1− 2 t)−n/^2 ; i.e.,Qis distributedχ^2 withndegrees of freedom.
In general, from Theorem 9.8.2, note how close the mgf of the quadratic form
Qis to the mgf of aχ^2 distribution. The next two theorems give conditions where
this is true.
Theorem 9.8.3.LetX′=(X 1 ,X 2 ,...,Xn)have aNn(μ,Σ)distribution, where
Σis positive definite. ThenQ=(X−μ)′Σ−^1 (X−μ)has aχ^2 (n)distribution.
Proof:Write the spectral decomposition ofΣasΣ=Γ′ΛΓ,whereΓis an orthog-
onal matrix andΛ=diag{λ 1 ,...,λn}is a diagonal matrix whose diagonal entries
are the eigenvalues ofΣ. BecauseΣis positive definite, allλi>0. Hencewecan
write
Σ−^1 =Γ′Λ−^1 Γ=Γ′Λ−^1 /^2 ΓΓ′Λ−^1 /^2 Γ,
whereΛ−^1 /^2 =diag{λ
− 1 / 2
1 ,...,λ
− 1 / 2
n }.Thuswehave
Q=
{
Λ−^1 /^2 Γ(X−μ)
}′
I
{
Λ−^1 /^2 Γ(X−μ)
}
.
But by Theorem 3.5.2, it is easy to show that the random vectorΛ−^1 /^2 Γ(X−μ)
has aNn( 0 ,I) distribution; hence,Qhas aχ^2 (n) distribution.
The remarkable fact that the random variableQin the last theorem isχ^2 (n)
stimulates a number of questions about quadratic forms in normally distributed