If $X_1,\ldots,X_n$ are iid $U(0,\theta)$, and $X(n) = \max(X_1,X_2,\ldots,X_n)$.
H0: $\theta < 1$
HA: $\theta \ge 1$
Reject Ho if $X(n) > c$ Accept H0 if $X(n) \le c$
- Find a constant $k$ such that the Type I error is 0.10.
Attempt: I know this is a one sided test, therefore a UMP exists. I know how to do this if the variables came from a normal distribution but am not sure how to do this if the sample comes from a uniform. Do I need to know the distribution of $X(n)$?
2. Determine the power function.
For the normal, I know $P(Z> c + ((\theta_0 - \theta)/(\sigma/\sqrt{n})$ for a simple hypothesis, but I am stuck in this case for a uniform and a one sided hypothesis.