# Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$.

Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: \theta_1 <0$ and $H_1: \theta_1 > 0$.

I think the test statistic should be $X_{(1)}$, the first order statistic, and it's distribution can be get from formula. But I wonder how to show that it's indeed UMPUT?

• If the minimum observed value is $1$ and the maximum is $1.2$ and there are a thousand observations, I'd feel fairly confident that $\theta_1>0$, but if the minimum observed value is $1$ and the maximum is $2000$ and they're scattered more-or-less uniformly in that range, then I would not feel particularly confident that $\theta_1>0$. So I suspect that some measure of dispersion is needed, and I wouldn't be at all surprised if it's the range. Certainly the pair $(X_{(1)},X_{(n)})$ is a sufficient statistic for this family of distributions. ${}\qquad{}$ – Michael Hardy Aug 10 '15 at 22:18
• Notice that within this context, if $X_{(1)}<0$, then you can be certain that $\theta_1<0$, but there is no observation on whose basis you can be certain that $\theta_1>0$. ${}\qquad{}$ – Michael Hardy Aug 10 '15 at 22:22
• You should be able to answer this by appealing to the Karlin-Rubin theorem. – dsaxton Aug 12 '15 at 4:13