Finding values for some test and power for uniform distribution

Question

I have the random sample $X_1, X_2, \dots, X_n$ drawn from the uniform distribution on $[\varphi, \varphi + 1]$. To test the null hypothesis $H_0 : \varphi = 0$ against the alternative hypothesis $H_1 : \varphi > 0$, we have the test

$$\text{Reject} \ H_0 \ \ \ \text{if} \ \ \ X_{(n)} \ge 1 \ \text{or} \ X_{(1)} \ge g,$$

where $g$ is a constant, $X_{(1)} = \min\{X_1, X_2, \dots, X_n\}, X_{(n)} = \max\{X_1, X_2, \dots, X_n\}$.

How do I find $n$ and $g$ so that the $0.05$ level will have power at least $0.8$ if $\varphi > 0.1$?

I think we have to begin by finding the CDF of $X_{(1)}$ and $X_{(n)}$, but I've only just started learning this stuff, so I really don't know what I'm supposed to do. I would appreciate it if any answers would please explain the reasoning behind the steps, so that I may better understand.

Answer 1

Hint to start you off:

For $x \in [\phi, \phi+1]$:

• $\mathbb P(X_{(n)} \le x) = \left(x-\phi\right)^n$ as you want all the values to be in $[\phi,x]$
• $\mathbb P(X_{(1)} \le x) = 1 -\left(\phi +1- x\right)^n$ as you want the complement of the event that all the values are in $(x,\phi+1]$