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## Neyman Pearson Theorem

Suppose is a random sample with joint density function . For simple and simple , the joint density function can be written as , , respectively. Alternatively, we could use the likelihood notation, , .

Theorem 9..1

In testing against , the critical region

is most powerful (where K 0).

[Or, in terms of likelihood, for a given , the test that maximizes the power at has rejection region determined by

Such a test will be most powerful for testing against .]

Proof. See Hogg and Craig. (Not required).

Example 9..5
Suppose represents a single observation from the probability density function given by

Find the most powerful (MP) test with significance level to test versus .

Solution. Since both and are simple, the previous Theorem can be applied to derive the test. Here

The form of the rejection region for the MP test is

Equivalently, or, since is a constant ( say), the critical region is .

The value of is determined by

So . So the rejection region is C . Among all tests for versus based on a sample of size 1 and , this test has smallest Type II error probability.
[ Note that the form of the test statistic and rejection region depends on both and . If is changed to , the MP test is based on and we reject in favour of if for some ].

Next: Uniformly Most Powerful (UMP) Up: Evaluation of and Construction Previous: Certain Best Tests   Contents
Bob Murison 2000-10-31