Consider two probability density functions on [0,1]:$f_0(x) = 1$, and $f_1(x) = 2x$. Among all the tests of the null hypothesis $H_0 : X \sim f_0(x)$ versus the alternative $X \sim f_1(x)$, with a significance level $\alpha$ = 0.10. How large can the power possibly be?
Any idea on how to find the power of this test?
Hint 1: Note that the hypothesis can be characterized in the following form: $$H_0 : \theta = 1, \quad \text{vs.} \quad H_1 : \theta = 2,$$ where $$X \sim \operatorname{Beta}(\theta,1), \quad f_X(x) = \theta x^{\theta-1}, \quad x \in [0,1].$$ So what we have here is a simple (point) hypothesis for the value of the parameter $\theta$ for a test of a fixed size $\alpha = 0.1$. What kind of test is uniformly most powerful in this situation? What theorem proves that such a test is UMP?
Hint 2: What is the test statistic for the UMP test? Can you calculate the rejection region for this test?
Hint 3: What is the probability of failing to reject $H_0$ when $H_1$ is true, under this test?