Ways to find a UMP test

Question

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?

Answer 1

(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).