Lecture 13. A) Test Optimality (cont.)

From Significant Statistics
Jump to navigation Jump to search

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.