I'm studying for my final exams and the subject of proof will basically test hypotheses, I will try to summarize here my doubts.

For found the UMP test the ways are

1) Use Neyman–Pearson lemma where the test is of the type $$H_0:\theta=\theta_0\space vs \space H_1:\theta=\theta_1$$ and the pdf is $f(x|\theta)$ with critical region $R$ is $x\in R$ if $\frac{f(x|\theta_1)}{f(x|\theta_0)}>k$ and $x\in R^c$ if $\frac{f(x|\theta_1)}{f(x|\theta_0)}<k$ for any $k\geq 0$ and $\alpha=P_{\theta_0}(X\in R)$

2)If $T$ is a sufficient statistic and the family of pdf's of $T$ have monotone likelihood-ratio then you can apply Karlin-Rubin theorem for test the hypotheses $$H_0:\theta\leq \theta_0\space vs\space > H_1:\theta>\theta_0$$ where the test is reject $H_0\Leftrightarrow > T>t_0$ and $\alpha=P_{\theta_0}(T>t_0)$

I know that Neyman-Pearson lemma can only be applied to simple hypothesis, but there is a "trick" to apply Neyman-Pearson lemma, where you can change the simple hyphotesis $$H_0:\theta=\theta_0\space vs\space H_1:\theta=\theta_1$$ to $$H_0':\theta=\theta_0\space vs \space H_1':\theta=\theta_1, \space\theta_1>\theta_0$$ $$H_0':\theta=\theta_0\space vs \space H_1':\theta=\theta_1, \space\theta_1<\theta_0$$ $$H_0':\theta=\theta_0\space vs \space H_1':\theta=\theta_1, \space\theta_1\neq\theta_0$$

Let's take an example for me to try to clarify what I mean by trick. Suppose I want a UMP test $$H_0:\theta=5\space vs\space H_1:\theta>5$$ I can not apply directly the Neyman-Pearson lemma in this case, then I did $$H_0:\theta=5\space vs\space H_1:\theta=\theta_1\space,\space \theta_1>5$$

these changes implies something in the critical region? In the size of test?

I can do this kind of manipulation in the null hypothesis?

Is there any other way to find a UMP test?

Could someone explain me especially how to get a UMP test for a density that does not belong to the exponential family?

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    $\begingroup$ "Can someone also explain me the difference between the likelihood ratio test and apply the Neyman–Pearson lemma?" - I don't understand what you're asking. The N-P Lemma says the LRT is UMP for two simple hypotheses. $\endgroup$ – Scortchi - Reinstate Monica Jun 26 '15 at 9:48
  • $\begingroup$ @Scortchi I had seen the relationship, but as my book does not say anything explicitly was in doubt. Is there any other way to find a UMP test? $\endgroup$ – user72621 Jun 26 '15 at 12:40
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    $\begingroup$ I just wanted to clarify what you were asking there. Another thing I want to clarify after your edit is exactly what these "tricks" are. (I was also going to point out that monotone likelihood ratios can be found outside the exponential family, but you must've realized that.) $\endgroup$ – Scortchi - Reinstate Monica Jun 26 '15 at 13:54
  • $\begingroup$ @Scortchi My main question is how to find UMP tests for composite hypotheses?I edited it to try to explain what is the trick. $\endgroup$ – user72621 Jun 26 '15 at 14:57
  • $\begingroup$ So the trick's that the UMP test for $H_0:\theta=5$ vs $H_1:\theta>5$ is the same as the UMP test for, say, $H_0:\theta=5$ vs $H_1:\theta=6$, which you know from the N-P lemma is the LRT? $\endgroup$ – Scortchi - Reinstate Monica Jun 28 '15 at 23:08

(1) Part & parcel of being a uniformly most powerful test for $H_0:\theta=\theta_0$ vs $H_1:\theta>\theta_0$ is being most powerful for $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1$ for whichever $\theta_1>\theta$ you choose. So the tests are exactly the same. (But there isn't always a UMP test for one-sided alternative hypotheses. Testing hypotheses about the location parameter of a Cauchy with known scale is a standard example.)

(2) The Karlin-Rubin theorem tells you that there is a UMP test for a one-sided alternative hypothesis, & how to form it, when the density (or mass) function of the sufficient statistic has a monotone likelihood ratio. There's no caveat that its distribution must belong to an exponential family; rather if it does belong to the (full) exponential family it will have monotone likelihood ratio. The hypergeometric distribution provides an example of a test statistic whose distribution does not belong to the exponential family & yet whose mass function has a monotone likelihood ratio.

(3) I don't know of general methods for finding UMP tests other than those you've described. As noted above, they don't always exist; then restricting your search to UMP unbiased tests or locally most powerful tests might be of interest, as might showing that a test under consideration is admissible (i.e. there's no other test with greater power under all versions of the alternative).

  • $\begingroup$ "if it does belong to the exponential family it will have monotone likelihood ratio" I think you need "full regular" exponential family. And Pfanzagl proved that if there is a UMP test for one-sided test regardless of sample size, then it must be a 1-dim exponential family, which is a well-known result in "Pfanzagl, Johann. "A characterization of the one parameter exponential family by existence of uniformly most powerful tests." Sankhyā: The Indian Journal of Statistics, Series A (1968): 147-156." $\endgroup$ – Henry.L Jan 29 '17 at 2:23

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