UMPU versus UMP

Question

I feel like this is a really basic question, but its been low-key bothering me. I understand that UMP tests are UMPU tests, but that not all UMPU tests are UMP. What is a specific example of a UMPU test that is not UMP? It seems like UMP tests are the ideal tests. Is my understanding correct?

Answer 1

I'm not so sure if you're still curious about this subject, but I happen to know one example. So, UMP test is basically a test that has the size of $\alpha$ and has the strongest power among the ones with the same size. Let's take a look at an example below. $$ \text{Find UMP test of $H_0: \mu = \mu_0, H_1: \mu > \mu_0 $ from }X_1, \dots, X_n \overset{i.i.d}{\sim} N(\mu, 1) $$ Let us be given an arbitrary $\mu_1 > \mu_0$, then by Neyman Pearson Lemma, the MP test is as below, $$ \phi(x) = \begin{cases} 1, \bar{x} - \mu_0 \geq z_{\alpha}/\sqrt{n}\\ 0, \bar{x} - \mu_0 < z_{\alpha}/\sqrt{n} \end{cases} $$ Fortunately, this test works for all $\mu$'s not just this single $\mu_1$. So, we can find UMP by Neyman Pearson Lemma when the MP test does not depend on the alternative parameter space. I recommend you try the same thing when $H_0(-): \mu \leq \mu_0$ with the same alternative hypothesis. The test result does not change. However, the result is not in 'two-sided test' where $H_0: \mu = \mu_0, H_1: \mu \neq \mu_0$. UMP Test does not exist in this one, but UMPU does. The conditions for UMPU is as below, $$ \mathbb{E}_{\theta_0}\phi^*(X)=\alpha, [\frac{d}{d\theta}\mathbb{E}_{\theta}\phi^*(X)]_{\theta=\theta_0} = 0 $$ $$ \forall \phi: \mathbb{E}_{\theta_0}\phi(X) = \alpha, [\frac{d}{d\theta}\mathbb{E}_{\theta}\phi(X)]_{\theta=\theta_0} = 0 \\ \mathbb{E}_{\theta_1}\phi^*(X) \geq \mathbb{E}_{\theta_1}\phi(X) $$

The UMP does not but the UMPU $\phi^*$ does exist in this case. More information about UMPU can be found in Testing Statistical Hypothesis 3rd ed. [E. L. Lehmann & J. P. Romano]