488 Optimal Tests of Hypothesesit was shown that no UMP test exists for this situation. If we restrict attention
to the class of unbiased tests (Definition 8.1.2), then a theory of best tests can be
constructed; see Lehmann (1986). For ourillustrative example, as Exercise 8.3.21
shows, the test based on the critical regionC 2 ={
|X−θ′ 1 |>√
θ 2
nzα/ 2}is unbiased. Then it follows from Lehmann that it is an UMP unbiased levelαtest.
In practice, though, the varianceθ 2 is unknown. In this case, theory for optimal
tests can be constructed using the concept of what are called conditional tests.
We do not pursue this any further in this text, but refer the interested reader to
Lehmann (1986).
Recall from Chapter 6 that the likelihood ratio tests (6.3.3) can be used to test
general hypotheses such as (8.3.1). While in general the exact null distribution of the
test statistic cannot be determined, under regularity condtions the likelihood ratio
test statistic is asymptoticallyχ^2 underH 0. Hence we can obtain an approximate
test in most situations. Although, there is no guarantee that likelihood ratio tests
are optimal, similar to tests based on the Neyman–Pearson Theorem, they are
based on a ratio of likelihood functions and, in many situations, are asymptotically
optimal.
In the example above on testing for the mean of a normal distribution, with
known variance, the likelihood ratio test is the same as the UMP unbiased test.
When the variance is unknown, the likelihood ratio test results in the one-sample
t-test as shown in Example 6.5.1 of Chapter 6. This is the same as the conditional
test discussed in Lehmann (1986).
In the remainder of this section, we present likelihood ratio tests for situations
when sampling from normal distributions.
8.3.1 Likelihood Ratio Tests for Testing Means of Normal
Distributions
In Example 6.5.1 of Chapter 6, we derived the likelihood ratio test for the one-
samplet-test to test for the mean of a normal distribution with unknown variance.
In the next example, we derive the likelihood ratio test for compairing the means
of two independent normal distributions. We then discuss the power functions for
both of these tests.Example 8.3.1.Let the independent random variablesXandYhave distributions
that areN(θ 1 ,θ 3 )andN(θ 2 ,θ 3 ), where the meansθ 1 andθ 2 and common varianceθ 3
are unknown. Then Ω ={(θ 1 ,θ 2 ,θ 3 ):−∞<θ 1 <∞,−∞<θ 2 <∞, 0 <θ 3 <∞}.
LetX 1 ,X 2 ,...,XnandY 1 ,Y 2 ,...,Ymdenote independent random samples from
these distributions. The hypothesisH 0 :θ 1 =θ 2 , unspecified, andθ 3 unspecified,
is to be tested against all alternatives. Thenω={(θ 1 ,θ 2 ,θ 3 ):−∞<θ 1 =θ 2 <
∞, 0 <θ 3 <∞}.HereX 1 ,X 2 ,...,Xn,Y 1 ,Y 2 ,...,Ymaren+m>2 mutually