i am trying to find the form of the uniformly most powerful test for a variable with a certain pdf.
the PDF is $f(x;\theta) = \theta \cdot (1 - x)^{\theta - 1}$ and $0 < x < 1$ where $\theta > 0$.
i need to find the form of the uniformly most powerful test for $H_0: \theta = 1$ against $H_A: \theta > 1$.
i'm not really sure how to go about solving this problem because my book doesn't have an example that looks like this.
the examples that i have seen suggest that i am supposed to use a ratio of likelihood functions, e.g., $\frac{L(2)}{L(5)}$, set the ratio less than or equal to some constant $k$ and then solve for the estimator in the hypothesis test. for a normal variable that would be $\bar x$ and for a binomial it would be $\frac{y}{n} = p$.
here i suppose it would be $\theta$, but i'm not sure how to proceed.
one thing i've tried is setting $\theta = 1$ and $\theta = 2$ and computing $\frac{L(1)}{L(2)} = \frac{1}{2 - 2x} \leq k$, then breaking this up with natural logs to get $\ln(1) - \ln(2 - 2x) \leq \ln(k)$, but then i don't know where to go from there.
any advice? thank you in advance.