# Hypothesis Testing under Uniform Distribution Question

Let $$\theta > 0$$ and $$X \sim \mathcal{U}[0, \theta]$$, i.e. $$X$$ is uniformly distributed on the interval $$[0, \theta]$$.

Assume that $$\theta$$ is unknown, but we can observe $$X$$. For given $$\theta_1$$, we want to test the hypothesis H0: $$\theta \geq \theta_1$$ against the alternative H1 : $$\theta < \theta_1$$. Consider the test which rejects H0, if and only if $$X < c$$. How should we choose $$c$$, as a function of $$\theta_1$$ and $$\alpha$$, to get a test with confidence level $$\alpha$$? Carefully justify your answer.

I struggle to understand how to approach this question/ how to carry it out? Any help would be much appreciated as this question is due in today. Thanks!!

• What is uniformly distributed? – the_candyman Feb 26 at 14:55
• @the_candyman Sorry, completely blanked out the first bit of the question, have edited the post to make it correct now. Should make sense. – James Debenham Feb 26 at 15:00

The significance level $$\alpha$$ corresponds to the maximum Type I error you are willing to accept for the test; that is to say, the erroneous conclusion to reject the null hypothesis when it is true: $$\Pr[\text{reject } H_0 \mid H_0 \text{ true}] \le \alpha.$$ Since you are already told that the criterion to reject $$H_0$$ is if $$X < c$$, and you are also told that the null hypothesis is $$H_0 : \theta \ge \theta_1$$, this becomes $$\Pr[X < c \mid \theta \ge \theta_1] \le \alpha.$$
1. For a fixed $$\theta$$, what is the conditional probability $$\Pr[X < c \mid \theta]$$?
2. As $$\theta$$ increases, does $$\Pr[X < c \mid \theta]$$ increase, or decrease, for a fixed $$c$$?
3. How does this inform the choice of $$\theta$$ under $$H_0$$, and $$c$$ as a function of $$\theta_1$$ and $$\alpha$$, to ensure a level $$\alpha$$ test?