Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.8. The Distributions of Certain Quadratic Forms 557

proof thatS^2 is an unbiased estimate ofσ^2. Note that the mean of the random

vectorXisμ 1 and that its variance–covariance matrix isσ^2 I. Based on Theorem

9.8.1, we find immediately that

E(S^2 )=

1

n− 1

{

tr

(

I−

1

n

J

)

σ^2 I+μ^2

(

1 ′ 1 −

1

n

1 ′ 11 ′ 1

)}

=σ^2.

The spectral decomposition of symmetric matrices proves quite useful in this part
of the chapter. As discussed around expression (3.5.8), a real symmetric matrixA
can be diagonalized as
A=Γ′ΛΓ, (9.8.8)
whereΛis the diagonal matrixΛ=diag(λ 1 ,...,λn),λ 1 ≥···≥λnare the eigen-
values ofA, and the columns ofΓ′=[v 1 ···vn] are the corresponding orthonormal
eigenvectors (i.e.,Γis an orthogonal matrix). Recall from linear algebra that the
rank ofAis the number of nonzero eigenvalues. Further, becauseΛis diagonal, we
can write this expression as

A=

∑n

i=1

λiviv′i. (9.8.9)

The R command to compute the spectral decomposition ofAissdc=eigen(amat),

whereamatis the R matrix forA. The eigenvalues and eigenvectors are in the

respective attributessdc$valuesandsdc$vectors. For normal random variables,

we make use of equation (9.8.9) to obtain the mgf of the quadratic formQin the

next theorem, Theorem 9.8.2.

Theorem 9.8.2.LetX′=(X 1 ,...,Xn),whereX 1 ,...,Xnare iidN(0,σ^2 ).Con-
sider the quadratic formQ=σ−^2 X′AXfor a symmetric matrixAof rankr≤n.
ThenQhas the moment generating function

M(t)=

∏r

i=1

(1− 2 tλi)−^1 /^2 =|I− 2 tA|−^1 /^2 , (9.8.10)

whereλ 1 ,...,λrare the nonzero eigenvalues ofA,|t|< 1 /(2λ∗), and the value of
λ∗is given byλ∗=max 1 ≤i≤r|λi|.

Proof: Write the spectral decomposition ofAas in expression (9.8.9). Since the
rank ofAisr,exactlyrof the eigenvalues are not 0. Denote the nonzero eigenvalues
byλ 1 ,...,λr.ThenwecanwriteQas

Q=

∑r

i=1

λi(σ−^1 vi′X)^2. (9.8.11)

LetΓ′ 1 =[v 1 ···vr] and define the r-dimensional random vectorWby W =
σ−^1 Γ 1 X.SinceXisNn( 0 ,σ^2 In)andΓ′ 1 Γ 1 =Ir, Theorem 3.5.2 shows thatW
has aNr( 0 ,Ir) distribution. In terms of theWi, we can write (9.8.11) as

Q=

∑r

i=1

λiWi^2. (9.8.12)