9.8. The Distributions of Certain Quadratic Forms 561(a)Use Theorem 9.8.1 to find theE(Q).(b)Justify the assertion thatQisχ^2 (2,6).9.8.4. SupposeX 1 ,...,Xnare independent random variables with the common
meanμbut with unequal variancesσi^2 =Var(Xi).
(a)Determine the variance ofX.(b)Determine the constantK so thatQ=K∑n
i=1(Xi−X)(^2) is an unbiased
estimate of the variance ofX.(Hint:Proceed as in Example 9.8.3.)
9.8.5.SupposeX 1 ,...,Xnare correlated random variables, with common meanμ
and varianceσ^2 but with correlationsρ(all correlations are the same).
(a)Determine the variance ofX.
(b)Determine the constantK so thatQ=K
∑n
i=1(Xi−X)
(^2) is an unbiased
estimate of the variance ofX.(Hint:Proceed as in Example 9.8.3.)
9.8.6.Fill in the details for expression (9.8.13).
9.8.7. For the trace operator defined in expression (9.8.1), prove the following
properties are true.
(a)IfAandBaren×nmatrices andaandbare scalars, then
tr (aA+bB)=atrA+btrB.
(b)IfAis ann×mmatrix,Bis anm×kmatrix, andCis ak×nmatrix, then
tr (ABC)=tr(BCA)=tr(CAB).
(c)IfAis a square matrix andΓis an orthogonal matrix, use the result of part
(a) to show that tr(Γ′AΓ)=trA.
(d)IfAis a real symmetric idempotent matrix, use the result of part (b) to prove
that the rank ofAis equal to trA.
9.8.8.LetA=[aij] be a real symmetric matrix. Prove that
∑
i
∑
ja
2
ijis equal to
the sum of the squares of the eigenvalues ofA.
Hint: IfΓis an orthogonal matrix, show that
∑
j
∑
ia
2
ij=tr(A
(^2) )=tr(Γ′A (^2) Γ)=
tr[(Γ′AΓ)(Γ′AΓ)].
9.8.9.SupposeXhas aNn(0,Σ) distribution, whereΣis positive definite. Let
Q=X′AXfor a symmetric matrixAwith rankr.ProveQhas aχ^2 (r) distribution
if and only ifAΣA=A.
Hint:WriteQas
Q=(Σ−^1 /^2 X)′Σ^1 /^2 AΣ^1 /^2 (Σ−^1 /^2 X),
whereΣ^1 /^2 =Γ′Λ^1 /^2 ΓandΣ=Γ′ΛΓis the spectral decomposition ofΣ.Then
use Theorem 9.8.4.
9.8.10.SupposeAis a real symmetric matrix. If the eigenvalues ofAare only 0s
and 1s then prove thatAis idempotent.