Page 451 - Probability and Statistical Inference

428 8. Tests of Hypotheses

One can use the test function ψ(X) for testing H versus too. In
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view of (8.5.7), we realize that ψ(X) has level α and so for θ < θ the two test
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functions must coincide. That is, ψ(X) corresponds to a UMP
level α test for H versus Next, by comparing the form of ψ(X)
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described in the statement of the theorem with ψ*(X) given in (8.5.3), we
can claim that ψ(X) also corresponds to a UMP level α test for H versus
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Hence the result follows.¾

8.5.3 Unbiased and UMP Unbiased Tests

It should be clear that one would prefer a UMP level a test to decide between
two hypotheses, but the problem is that there may not exist such a test. Since,
the whole class of level α tests may be very large in the first place, one may
be tempted to look at a sub-class of competing tests and then derive the best
test in that class. There are, however, different ways to consider smaller
classes of tests. We restrict our attention to the unbiased class of tests.
Definition 8.5.1 Consider testing H : θ ∈ Θ versus H : θ ∈ Θ where Θ ,
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0
1
0
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Θ are subsets of Θ and Θ , Θ are disjoint. A level a test is called unbiased if
1
0
1
Q(θ), the power of the test, that is the probability of rejecting H , is at least α
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whenever θ ∈ Θ .
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Thus, a test with its test function ψ(X) would be called unbiased level a
provided the following two conditions hold:
The properties listed in (8.5.9) state that an unbiased test rejects the null
hypothesis more frequently when the alternative hypothesis is true than in a
situation when the null hypothesis is true. This is a fairly minimal demand on
any reasonable test used in practice.
Next, one may set out to locate the UMP level a test within the class of
level a unbiased tests. Such a test is called the uniformly most powerful unbi-
ased (UMPU) level a test. An elaborate theory of unbiased tests was given by
Neyman as well as Neyman and Pearson in a series of papers. The readers
will find a wealth of information in Lehmann (1986).
When testing a simple null hypothesis H : θ = θ versus a simple alterna-
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tive H : θ = θ , the Neyman-Pearson Lemma came up with the MP level a test
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1
implemented by the test function ψ(x) defined by (8.3.2). Now, consider
another test function ψ*(x) ≡ α, that is this randomized test rejects H with
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n
probability a whatever be the observed data x ∈ χ .