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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.

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    $\begingroup$ please give exact descriptions for $\mathcal C$ and $\mathcal C^c$ ... etc. It would prevent me from guessing. NO, not all LRT are UMP. Most of the times LRT are UMP same, and this is true when MLR holds (that's why MLR property is an important one!). A very typical example, two-sided two sample t-test ($H_0: \Delta \mu=0 \text{ vs } H_1: \Delta \mu\neq0$). $\endgroup$
    – suncoolsu
    Aug 4, 2011 at 7:36
  • $\begingroup$ $\mathcal C$ is the critical (rejection) region and $\mathcal C^c$ its complement (acceptance region). So the first statement seems to be saying that for a UMP test, the critical value for the test statistic should be a linear function of the test statistic evaluated under the null hypothesis? I can't think of any statistical test that doesn't fit this definition, so I'm wondering what I'm missing. $\endgroup$
    – f1r3br4nd
    Aug 4, 2011 at 13:22
  • $\begingroup$ So, given a particular null distribution, if there is a monotone likelihood ratio (either because it's a one-tailed test or the distribution itself is monotone) then the LRT for that distribution is also the UMP, is that right? Are there UMP tests that are not LRT? Thanks. $\endgroup$
    – f1r3br4nd
    Aug 4, 2011 at 13:30

2 Answers 2

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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.

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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?

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