Confidence interval for $X \sim U(0,\frac{1}{\theta})$

Question

If $X \sim U(0,\frac{1}{\theta})$ and $Y= \theta X_{(n)}$ where $X_{(n)}=\max \{X_1, X_2, ..., X_n \}$.

$F_{Y}(y)=y^n \quad \ f_{Y}(y)=n z^{n-1}$

And then $P(a<\theta X_{(n)}<b)=1 - \alpha \implies P( \frac{a}{X_{(n)}} < \theta < \frac{b}{X_{(n)}})=1 - \alpha$.

What happens when $X_{(n)}=0$?

Answer 1

Note that because $X_{(n)}$ has a density, we have $P(X_{(n)} = 0) = 0$. Since the probability is zero, we may ignore this case, so that the interval you obtained is perfectly fine.