6.5. Multiparameter Case: Testing 397Following Example 6.4.1, it is easy to show that under the reduced parameter space
ω,̂σ 02 =(1/n)∑n
i=1(Xi−μ^0 )(^2). Thus the maximum value of the likelihood function
underωis
L(ω̂)=
1
(2π)n/^2
1
(σ̂^20 )n/^2
exp{−(n/2)}. (6.5.7)
The likelihood ratio test statistic is the ratio ofL(ω̂)toL(Ω); i.e,̂
Λ=
(∑n
i=1(Xi−X)
2
∑n
i=1(Xi−μ^0 )
2
)n/ 2
. (6.5.8)
The likelihood ratio test rejectsH 0 if Λ≤c, but this is equivalent to rejectingH 0
if Λ−^2 /n≥c′. Next, consider the identity∑ni=1(Xi−μ 0 )^2 =∑ni=1(Xi−X)^2 +n(X−μ 0 )^2. (6.5.9)Substituting (6.5.9) for∑n
i=1(Xi−μ^0 )(^2) , after simplification, the test becomes reject
H 0 if
1+
n(X−μ 0 )^2
∑n
i=1(Xi−X)
2 ≥c
′,
or equivalently, rejectH 0 if
⎧
⎨
⎩
√
n(X−μ 0 )
√∑
n
i=1(Xi−X)
(^2) /(n−1)
⎫
⎬
⎭
2
≥c′′=(c′−1)(n−1).
LetTdenote the expression within braces on the left side of this inequality. Then
the decision rule is equivalent to
RejectH 0 in favor ofH 1 if|T|≥c∗, (6.5.10)
whereα =PH 0 [|T|≥c∗]. Of course, this is the two-sided version of thet-test
presented in Example 4.5.4. If we takecto betα/ 2 ,n− 1 , the upperα/2-critical value
of at-distribution withn−1 degrees of freedom, then our test has exact levelα.
The power function for this test is discussed in Section 8.3.
As discussed in Example 4.2.1, the R call to computetist.test(x,mu=mu0),
where the vectorxcontains the sample and the scalarmu0isμ 0. It also computes
thet-confidence interval forμ.
Other examples of likelihood ratio tests for normal distributions can be found
in the exercises.
We are not always as fortunate as in Example 6.5.1 to obtain the likelihood
ratio test in a simple form. Often it is difficult or perhaps impossible to obtain its
finite sample distribution. But, as the next theorem shows, we can always obtain
an asymptotic test based on it.