Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

566 Inferences About Normal Linear Models

This section concludes with a proof of a frequently quoted theorem due to
Cochran.

Theorem 9.9.3(Cochran).LetX 1 ,X 2 ,...,Xndenote a random sample from a
distribution that isN(0,σ^2 ). Let the sum of the squares of these observations be
written in the form
∑n

1

Xi^2 =Q 1 +Q 2 +···+Qk,

whereQj is a quadratic form inX 1 ,X 2 ,...,Xn,withmatrixAj that has rank
rj,j=1, 2 ,...,k. The random variablesQ 1 ,Q 2 ,...,Qk are independent and
Qj/σ^2 isχ^2 (rj),j=1, 2 ,...,k,ifandonlyif

∑k

1 rj=n.

Proof. First assume the two conditions

∑k

1 rj=nand

∑n

1 X

2

i =

∑k
1 Qj to be
satisfied. The latter equation implies thatI=A 1 +A 2 +···+Ak.LetBi=I−Ai;
that is,Biis the sum of the matricesA 1 ,...,Akexclusive ofAi.LetRidenote
the rank ofBi. Since the rank of the sum of several matrices is less than or equal
to the sum of the ranks, we haveRi≤

∑k
1 rj−ri=n−ri. However,I=Ai+Bi,
so thatn≤ri+Riandn−ri≤Ri. HenceRi=n−ri. The eigenvalues ofBiare
the roots of the equation|Bi−λI|=0. SinceBi=I−Ai, this equation can be
written as|I−Ai−λI|=0. Thuswehave|Ai−(1−λ)I|= 0. But each root of
the last equation is 1 minus an eigenvalue ofAi.SinceBihas exactlyn−Ri=ri
eigenvalues that are zero, thenAihas exactlyrieigenvalues that are equal to 1.
However,riis the rank ofAi.Thuseachoftherinonzero eigenvalues ofAiis 1.
That is,A^2 i=Aiand thusQi/σ^2 has aχ^2 (ri), fori=1, 2 ,...,k. In accordance
with Theorem 9.9.2, the random variablesQ 1 ,Q 2 ,...,Qkare independent.
To complete the proof of Theorem 9.9.3, take
∑n

1

Xi^2 =Q 1 +Q 2 +···+Qk,

letQ 1 ,Q 2 ,...,Qkbe independent, and letQj/σ^2 beχ^2 (rj),j=1, 2 ,...,k.Then
∑k
1 Qj/σ

(^2) isχ (^2) (∑k
1 rj). But
∑k
1 Qj/σ
(^2) =∑n
1 X
2
i/σ
(^2) isχ (^2) (n). Thus∑k
1 rj=n
and the proof is complete.
EXERCISES
9.9.1.LetX 1 ,X 2 ,X 3 be a random sample from the normal distributionN(0,σ^2 ).
Are the quadratic formsX 12 +3X 1 X 2 +X 22 +X 1 X 3 +X 32 andX 12 − 2 X 1 X 2 +^23 X 22 −
2 X 1 X 2 −X 32 independent or dependent?
9.9.2.LetX 1 ,X 2 ,...,Xndenote a random sample of sizenfrom a distribution
that isN(0,σ^2 ). Prove that
∑n
1 X
2
iand every quadratic form, that is nonidentically
zero inX 1 ,X 2 ,...,Xn, are dependent.
9.9.3. LetX 1 ,X 2 ,X 3 ,X 4 denote a random sample of size 4 from a distribution
that isN(0,σ^2 ). LetY=
∑ 4
1 aiXi,wherea^1 ,a^2 ,a^3 ,anda^4 are real constants. If
Y^2 andQ=X 1 X 2 −X 3 X 4 are independent, determinea 1 ,a 2 ,a 3 ,anda 4.