Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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 <

1. 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.