9.7. A Test of Independence 553ThusT 2 ,givenX 1 =x 1 ,...,Xn=xn, has a conditionalt-distribution withn− 2
degrees of freedom. Note that the pdf, sayg(t), of thist-distribution does not depend
uponx 1 ,x 2 ,...,xn.NowthejointpdfofX 1 ,X 2 ,...,XnandR
√
n− 2 /√
1 −R^2 ,
whereRis given by expression (9.7.1), is the product ofg(t)andthejointpdfof
X 1 ,...,Xn. Integration onx 1 ,...,xnyields the marginal pdf ofR
√
n− 2 /√
1 −R^2 ;
becauseg(t) does not depend uponx 1 ,x 2 ,...,xn, it is obvious that this marginal pdf
isg(t), the conditional pdf ofR
√
n− 2 /√
1 −R^2. The change-of-variable technique
can now be used to find the pdf ofR.
Remark 9.7.1.SinceRhas, whenρ= 0, a conditional distribution that does not
depend uponx 1 ,x 2 ,...,xn(and hence that conditional distribution is, in fact, the
marginal distribution ofR), we have the remarkable fact thatRis independent of
X 1 ,X 2 ,...,Xn. It follows thatRis independent ofevery functionofX 1 ,X 2 ,...,Xn
alone, that is, a function that does not depend upon anyYi. In like manner,Ris
independent of every function ofY 1 ,Y 2 ,...,Ynalone. Moreover, a careful review of
the argument reveals that nowhere did we use the fact thatXhas a normal marginal
distribution. Thus, ifXandYare independent, and ifYhas a normal distribution,
thenRhas the same conditional distribution whatever the distribution ofX,subject
to the condition
∑n
1 (xi−x)(^2) >0. Moreover, ifP[∑n
1 (Xi−X)
(^2) >0] = 1, thenR
has the same marginal distribution whatever the distribution ofX.
If we writeT=R
√
n− 2 /
√
1 −R^2 ,whereThas at-distribution withn− 2 > 0
degrees of freedom, it is easy to show by the change-of-variable technique (Exercise
9.7.4) that the pdf ofRis given by
h(r)=
{
Γ[(n−1)/2]
Γ(^12 )Γ(n−2)/2(n−4)/ (^2) − 1 <r< 1
0elsewhere.
(9.7.4)
We have now solved the problem of the distribution ofR,whenρ=0andn>2,
or perhaps more conveniently, that ofR
√
n− 2 /
√
1 −R^2. The likelihood ratio test
of the hypothesisH 0 :ρ= 0 against all alternativesH 1 :ρ = 0 may be based either
on the statisticRor on the statisticR
√
n− 2 /
√
1 −R^2 =T, although the latter is
easier to use. Therefore, a levelαtest is to rejectH 0 :ρ=0if|T|≥tα/ 2 ,n− 2.
Remark 9.7.2. It is possible to obtain an approximate test of sizeαby using the
fact that
W=
1
2
log
(
1+R
1 −R
)
has an approximate normal distribution with mean^12 log[(1 +ρ)/(1−ρ)] and with
variance 1/(n−3). We accept this statement without proof. Thus a test ofH 0 :
ρ= 0 can be based on the statistic
Z=
1
2 log[(1 +R)/(1−R)]−
1
√^2 log[(1 +ρ)/(1−ρ)]
1 /(n−3)
, (9.7.5)