Practical definition of a UMP test?

Question

I'm trying to make sense of these two statements about UMP (uniformly most powerful) tests:

1. If $g(t\mid\theta)~$ is a UMP then $~g(t\mid\theta_1)>k g(t\mid\theta_0)~\forall~t\in C$ and $g(t\mid\theta_1)<k g(t\mid\theta_0)~\forall~t\in C^c$

2. For $\mathbf X~i.i.d.~f(x\mid\theta): \theta\in\Omega\subset\mathcal R$ if $f(x\mid\theta)$ has an MLR in $T(x)$ and any $k$, a test that rejects $H_0 \iff T>k~$ is a UMP test of size $\alpha$ with $\alpha=P_{\theta_0}(T>k)$

Why are these statements not tautological? How are they not the definition of any one-tailed statistical test? What would be an example of a one-tailed statistical test that isn't a UMP?

Moreover, what is the relationship between LRT and UMP tests? I'm reviewing old exams where sometimes a question asks for an LRT and sometimes a UMP... Aren't all simple LRT tests UMP?

Thanks.

Answer 1

I'm having a hard time figuring out your notation; you might want to define things a little more so that we don't have to fill in the blanks. I assume $f$ and $g$ are supposed to be the pdf or pmf of something or other. I assume by a "one-tailed test" you mean a test of the hypothesis $H_0: \theta \le \theta_0$ against $H_1: \theta > \theta_0$ for some $\theta_0$. The first statement references a $\theta_0$ and $\theta_1$ which suggests to me that it is actually concerning the Neyman-Pearson lemma and not UMP tests as such. Here are some thoughts.

1. There are UMP tests that are not LRTs. There are even MP tests for testing $H_0: \theta = \theta_0$ against $H_1: \theta = \theta_1$ that are not LRTs. Essentially this is because LRTs do not allow for randomization, so for discrete rvs there may not exist a LRT of a pre-specified size. However, LRTs are MP for testing a simple null against a simple alternative and are UMP for the one-sided hypothesis provided that we have MLR; it just isn't the other way around.

2. No, not all tests of a one-sided hypothesis are of this form. For example, we might reject if $0 = 0$ and accept otherwise. Restricting to a "one-sided" test (whatever that means) we still might use tests like reject if $X_1 > k$ and accept otherwise, which seems one-sided enough, but is clearly not UMP since it doesn't use all the data.

3. It may be helpful to note that a LRT always exists, whereas in most situations the UMP test does not exist; this may explain why sometimes you are asked to derive a LRT (since you can cook them up for complicated situations fairly easily) sometimes and a UMP test others. UMP'ness is a rather special thing.

Answer 2

I'm not sure but here's what I've been able to make of this type of stuff so far:

1. According to the Neyman-Pearson lemma, if you compare the likelihood of the alternative hypothesis to a critical value times the likelihood of the null hypothesis (or, equivalently, the ratio of the alternative and null likelihoods to the critical value) then that's an MP. For one-tailed tests, this becomes a UMP. This generalizes to the log ratio test.

2. According to the Karlin-Rubin theorem, if the likelihood ratio is monotone as a function of a complete statistic, then you can just use the complete statistic itself as the test statistic and choose a critical value based on its distribution. This is also a UMP if it's a one-tailed test.

What do the rest of you think, is that right?