Lecture 13. A) Test Optimality (cont.)

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Test Optimality (cont.)

Optimality for tests with simple null and alternative hypotheses has been established by the Neyman-Pearson lemma.

For one-sided tests, the Neyman-Pearson lemma is also useful:

Consider the testing problem [math]H_{0}:\theta=\theta_{0}[/math] vs. [math]H_{1}:\theta\gt \theta_{0}[/math]. We consider each value of the alternative hypothesis [math]\theta_{1}\gt \theta_{0}[/math]. Each value yields the simple test [math]H_{0}:\theta=\theta_{0}[/math] vs. [math]H_{1}^{*}:\theta=\theta_{1}[/math]. We then use the Neyman-Pearson lemma at each value of [math]\theta_{1}\gt \theta_{0}[/math]. If all values of [math]\theta_{1}[/math] yield the same UMP level [math]\alpha[/math] test, then we have found the unique UMP level [math]\alpha[/math] test.

The lemma implies that:

  • If these tests coincide for all values of [math]\theta_{1}\gt \theta_{0}[/math], then the UMP test is [math]\frac{f\left(\left.x\right|\theta_{1}\right)}{f\left(\left.x\right|\theta_{0}\right)}\gt k[/math].
  • If the tests do not coincide for all values of [math]\theta_{1}\gt \theta_{0}[/math], then no UMP level [math]\alpha[/math] test exists.

When we say that the tests coincide for all values of [math]\theta_{1}\gt \theta_{0}[/math], we mean that no matter the specific value of [math]\theta_{1}[/math], we will obtain the same exact test. This will be clearer with the following example.