Page 531 - Probability and Statistical Inference

508 11. Likehood Ratio and Other Tests

Note that small values of Λ are associated with the small values of
relative to If the best evidence in favor of the null hypothesis ap-
pears weak, then the null hypothesis is rejected. That is, we reject H for
0
significantly small values of Λ.
It is easy to see that one must have 0 < Λ < 1 because in the definition of
Λ, the supremum in the numerator (denominator) is taken over a smaller
(larger) set Θ (Θ). The cut-off number k ∈ (0, 1) has to be chosen in such a
0
way that the LR test from (11.1.3) has the required level α.
For simplicity, we will handle only a special kind of null hypothesis. Let us
test

where is a known and fixed value of the (sub-) parameter θ . One may be
1
tempted to say that the hypothesis H is a simple null hypothesis. But, actually
0
it may not be so. Even though H specifies a fixed value for a single compo-
0
nent of θ, observe that the other components of ? remain unknown and arbi-
trary.
How should we evaluate First, in the expression of L(θ), we
must plug in the value in the place of . Then, we maximize the likelihood
function L( , θ , ..., θ ) with respect to the (sub-) parameters θ , ..., θ by
2
2
p
p
substituting their respective MLE’s when we know that θ = On the other
1
hand, the is found by plugging in the MLE’s of all the components
θ , ..., θ in the likelihood function. In the following sections, we highlight
p
1
these step by step derivations in a variety of situations.
Section 11.2 introduces LR tests for the mean and variance of a normal
population. In Section 11.3, we discuss LR tests for comparing the means
and variances of two independent normal populations. In Section 11.4, under
the assumption of bivariate normality, test procedures are given for the popu-
lation correlation coefficient ? and for comparing the means as well as vari-
ances.
11.2 One-Sample Problems

We focus on a single normal population and some LR tests associated
with it. With fixed α ∈ (0,1), first a level α LR test is derived for a speci-
fied population mean against the two-sided alternative hypothesis, and come
up with the customary two-sided Z-test (t-test) when the population vari-
ance is assumed known (unknown). Next, we obtain a level α LR test for