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 $X(n) \le c$

1. 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.

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Who or what gave you the idea that all one-sided hypotheses admit a UMP test? – cardinal Nov 8 '11 at 1:58

Notice that $\max\{X_1,\ldots,X_n\}\le x$ precisely if $(X_1\le x\text{ and}\ldots\text{and }X_n\le x)$, and the probability of that can be found by using independence, i.e. by multiplying. Remember that $\Pr(X_1\le x)=x/\theta$ That gives you the cumulative distribution function of the maximum.
Assuming the null hypothesis is true, you want the probability to be $0.1$ that $\max > c$. So you know that $\Pr(\max > c) = 1 - (c/\theta)^n$ if $c < \theta$ and $=1$ if $c>\theta$. So $1-(c/1)^n = 0.1$ if $c=\text{what}$? (BTW, if the observed sample max is more than 1, that you can be certain that the null hypothesis is false.) – Michael Hardy Nov 9 '11 at 4:09