# Ways to find a UMP test

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?

• "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. Commented Jun 26, 2015 at 9:48
• @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?
– user72621
Commented Jun 26, 2015 at 12:40
• 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.) Commented Jun 26, 2015 at 13:54
• @Scortchi My main question is how to find UMP tests for composite hypotheses?I edited it to try to explain what is the trick.
– user72621
Commented Jun 26, 2015 at 14:57
• 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? Commented Jun 28, 2015 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.)
• @Henry.L: Thanks, & sorry for forgetting to reply. The paper went over my head rather; but I do understand that (almost) only one-parameter full exponential families will have a scalar sufficient statistic for i.i.d. samples regardless of sample size, & that their likelihood ratio is a monotone function of this parameter. The exception is of course families whose parameter determines the support - for e.g. a uniform distributions on 0 to $\theta$ the sample maximum is sufficient & the likelihood ratio monotone. ... Commented Jan 22 at 14:18
• ... More importantly perhaps, we're often looking for U.M.P. similar or U.M.P. invariant tests. The one-sample t-test is a familiar example: the sample of $n$ is from a 2-dim exponential family with a 2-dim sufficient statistic yet what matters is the sample of 1 from a 1-dim non-exponential family (the non-central t-distribution with $n-1$ degrees of freedom) with a 1-dim sufficient statistic & monotone likelihood ratio. Fisher's Exact Test is another one. Commented Jan 22 at 14:21