Constructing a Confidence Interval for $\theta$ where $X_1, ..., X_8$ iid $U(0, \theta)$ where $\theta \ge 0$

Question

My homework question requires me to construct a 90% confidence for $\theta \ge 0$ where $X_1, ..., X_8$ iid $U(0, \theta)$ and the $8$ data values are given. However, I am not sure where to start with this. The question suggests using $\frac{X_{(n)}}{\theta}$ as a pivot variable (where $X_{(n)}=$ max{$X_1, ..., X_n$}), and I already have used $P(a \le \frac{X_{(n)}}{\theta} \le b)=0.9$ to get $a=0.05^{\frac{1}{n}}$ and $b=0.95^{\frac{1}{n}}$. I recognise that the solutions for $a$ and $b$ are not unique, however I just set each equal to the 5%-tile and 95%-tile respectively. I also know that I would let $n=8$ since there are $8$ observations, but am unsure of how to use this.

Answer 1

Using the hint you got, the pivotal quantity is $Y=\frac{X_{(n)}}{\theta}$

You surely know that

$f_Y(y)=8y^7\mathbb{1}_{(0;1)}(y)$

thus the optimum confidence interval is $[y;1]$

concluding:

$$X_{(8)}\leq \theta \leq \frac{X_{(8)}}{\sqrt[8]{0.1}}$$

EDIT:

the density of your pivotal quantity is the following

It is evident that there are $\infty$ way to find the extremes of a CI at $(1-\alpha)\%$ but in this case the most reasonable solution is to minimize the range of your CI so the interval is of the form $[a;1]$