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8. Tests of Hypotheses 413

should verify some of the entries in this table by using the expressions given
in (8.3.23). !

A MP α level a test always depends on (jointly) sufficient statistics.
In each example, the reader has noticed that the MP level α test always de-
pended only on the (jointly) sufficient statistics. This is not a coincidence.
Suppose that T = T(X , ..., X ) is a (jointly) sufficient statistic for θ. By the
1
n
Neyman factorization (Theorem 6.2.1), we can split the likelihood function as
follows:

where h(X) does not involve θ. Next, recall that the MP test rejects H : θ = θ 0
0
in favor of accepting H : θ = θ for large values of L(X; θ )/L(X; θ ). But, in
1
0
1
1
view of the factorization in (8.3.24), we can write
which implies that the MP test rejects H : θ = θ in favor of accepting H : θ
0
1
0
= θ , for large values of g(T(x); θ ))/g(T(x); θ )). Thus, it should not surprise
1
1
0
anyone to see that the MP tests in all the examples ultimately depended on the
(jointly) sufficient statistic T.
8.3.2 Applications: No Parameters Are Involved
In the statement of the Neyman-Pearson Lemma, we assumed that θ was a
real valued parameter and the common pmf or pdf was indexed by θ. We
mentioned earlier that this assumption was not really crucial. It is essential to
have a known and unique likelihood function under both H , H . In other
0
1
words, as long as H , H are both simple hypothesis, the Neyman-Pearson
0
1
Lemma will provide the MP level α test explicitly Some examples follow.
Example 8.3.8 Suppose that X is an observable random variable with its
pdf given by f(x), x ∈ ℜ. Consider two functions defined as follows:

We wish to determine the MP level α test for

In view of the Remark 8.3.1, the MP test will have the following form: