# Is the $p-$value uniformly distributed in this case?

Let $$(Ω, A,P)$$ be a statistical model, $$H_{0} = \{P_{0}\}\subseteq P$$ a simple null hypothesis, and $$H_{1} = \{P_{1}\} ⊂ P$$ a different simple alternative, so that $$P_{1}$$ with respect to $$P_{0}$$ has a density $$L = dP_{1}/dP_{0}$$. We assume the distribution $$\mathcal{L}_{P_{0}}(L)$$ of the Likelihood-Quotient $$L$$ under $$P_{0}$$ has a density $$f$$ w.r.t. the Lebesgue-measure. Let $$p : Ω → \mathbb R$$ be the $$p-$$value of the Likelihood-Quotient-Test (Neyman-Pearson-Test) for both Hypotheses. Which of the following statements $$A-E$$ is definitely true, and why? Only one is true:

$$A$$ The distribution $$L_{P_{0}}(p)$$ of $$p$$-value w.r.t. the null hypothesis has the density $$f$$ w.r.t the Lebesgue Measure.

$$B$$ The distribution $$L_{P_{1}}(p)$$ of $$p$$-value w.r.t. the alternative hypothesis has the density $$f$$ w.r.t the Lebesgue Measure.

$$C$$ The $$p-$$value is uniformly distributed w.r.t the null hypothesis on the unit interval

$$D$$ The $$p-$$value is uniformly distributed w.r.t the alternative hypothesis on the unit interval

$$E$$ The statements $$A-D$$ are incorrect.

I am new to statistics. But we do not have any critical region $$V$$ that is given. I know that the Likelihood Quotient Test is given by the critical region $$V$$, where $$\{ \frac{dP_{1}}{dP_{0}}>c\}\subseteq V \subseteq \{ \frac{dP_{1}}{dP_{0}}\geq c\}$$. Furthermore, the $$p-$$value is based on the infimum of significance levels $$a_{V}$$. I do not see how any of this may help me prove that statements $$A-E$$. Any ideas?

• See stats.stackexchange.com/questions/435833 and stats.stackexchange.com/questions/424169 for more elementary treatments of this result and let them guide you.
– whuber
Feb 13, 2020 at 20:05
• I do not see how I could adapt those to my case. What is the test statistic in my case? the c? Feb 13, 2020 at 22:24
• They don't need to be adapted: they are your case, merely framed a little differently.
– whuber
Feb 13, 2020 at 22:27