8.3. Likelihood Ratio Tests 491
and
m
(
Y−
nX+mY
n+m
) 2
=
n^2 m
(n+m)^2
(X−Y)^2.
Hence the random variable defined by Λ^2 /(n+m)may be written
∑n
1
(Xi−X)^2 +
∑m
1
(Yi−Y)^2
∑n
1
(Xi−X)^2 +
∑m
1
(Yi−Y)^2 +[nm/(n+m)](X−Y)^2
=
1
1+
[nm/(n+m)](X−Y)^2
∑n
1
(Xi−X)^2 +
∑m
1
(Yi−Y)^2
.
If the hypothesisH 0 :θ 1 =θ 2 is true, the random variable
T=
√
nm
n+m
(X−Y)
{
(n+m−2)−^1
[n
∑
1
(Xi−X)^2 +
∑m
1
(Yi−Y)^2
]}− 1 / 2
(8.3.4)
has, in accordance with Section 3.6, at-distribution withn+m−2 degrees of
freedom. Thus the random variable defined by Λ^2 /(n+m)is
n+m− 2
(n+m−2) +T^2
.
The test ofH 0 against all alternatives may then be based on at-distribution with
n+m−2 degrees of freedom.
The likelihood ratio principle calls for the rejection ofH 0 if and only if Λ≤λ 0 <
- Thus the significance level of the test is
α=PH 0 [Λ(X 1 ,...,Xn,Y 1 ,...,Ym)≤λ 0 ].
However, Λ(X 1 ,...,Xn,Y 1 ,...,Ym)≤λ 0 is equivalent to|T|≥c,andso
α=P(|T|≥c;H 0 ).
For given values ofnandm,thenumbercis is easily computed. In R,c=qt(1−
α/ 2 ,n+m−2). ThenH 0 is rejected at a significance levelαif and only if|t|≥c,
wheretis the observed value ofT. If, for instance,n=10,m=6,andα=0.05,
thenc=qt(0. 975 ,14) = 2.1448.
For this last example as well as the one-samplet-test derived in Example 6.5.1, it
was found that the likelihood ratio test could be based on a statistic that, when the
hypothesisH 0 is true, has at-distribution. To help us compute the power functions
of these tests at parameter points other than those described by the hypothesisH 0 ,
we turn to the following definition.