552 Inferences About Normal Linear Models

bivariate normal distribution; that is, the joint pdf of these 2nrandom variables is

given by

f(x 1 ,y 1 )f(x 2 ,y 2 )···f(xn,yn).

Although it is fairly difficult to show, the statistic that is defined by the likelihood
ratio Λ is a function of the statistic, which is the mle ofρ,namely,

R=

∑n

√ i=1(Xi−X)(Yi−Y)

∑n

i=1(Xi−X)

2 ∑n

i=1(Yi−Y)

2

. (9.7.1)

This statisticRis called the samplecorrelation coefficientof the random sam-
ple. Following the discussion after expression (5.4.5), the statisticRis a consistent
estimate ofρ; see Exercise 9.7.5. The likelihood ratio principle, which calls for the
rejection ofH 0 if Λ≤λ 0 , is equivalent to the computed value of|R|≥c.That
is, if the absolute value of the correlation coefficient of the sample is too large, we
reject the hypothesis that the correlation coefficient of the distribution is equal to
zero. To determine a value ofcfor a satisfactory significance level, it is necessary
to obtain the distribution ofR, or a function ofR,whenH 0 is true, as we outline
next.

∑LetX^1 =x^1 ,X^2 =x^2 ,...,Xn=xn,n>2, wherex^1 ,x^2 ,...,xn andx=
n
1 xi/nare fixed numbers such that

∑n

1 (xi−x)

(^2) >0. Consider the conditional pdf
ofY 1 ,Y 2 ,...,Yngiven thatX 1 =x 1 ,X 2 =x 2 ,...,Xn=xn. BecauseY 1 ,Y 2 ,...,Yn
are independent and, withρ = 0, are also independent ofX 1 ,X 2 ,...,Xn,this
conditional pdf is given by
(
1
√
2 πσ 2
)n
exp
{
−
1
2 σ^22
∑n
1
(yi−μ 2 )^2
}
.
LetRcbe the correlation coefficient, givenX 1 =x 1 ,X 2 =x 2 ,...,Xn=xn,sothat
Rc
√√
√
√
∑n
i=1
(Yi−Y)^2
√√
√
√
∑n
i=1
(xi−x)^2

∑n
i=1
(xi−x)(Yi−Y)
∑n
i=1
(xi−x)^2

∑n
i=1
(xi−x)Yi
∑n
i=1
(xi−x)^2
(9.7.2)
isβˆ, expression (9.6.5) of Section 9.6. Conditionally the mean ofYiisμ 2 ; i.e., a
constant. So here expression (9.7.2) has expectation 0 which implies thatE(Rc)=0.
Next consider thet-ratio ofβˆgiven byT 2 of expression (9.6.10) of Section 9.6. In
this notationT 2 can be expressed as
T 2 =
Rc
√∑
(Yi−Y)^2 /
√∑
(xi−x)^2
√
Pn
i=1
n
Yi−Y−
h
Rc
qPn
j=1(Yj−Y)^2 /
√Pn
j=1(xj−x)^2
i
(xi−x)
o 2
(n−2)Pnj=1(xj−x)^2

Rc
√
n− 2
√
1 −R^2 c
.
(9.7.3)