.

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

The lemma implies that:

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

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