I'm doing Exercise 6.26 in Casella and Berger's Statistical Inference, and I'm trying to prove the following:
"Use Theorem 6.6.5 to establish that, given a sample $X_1,...,X_n$, the maximum order statistic $X_{(n)}$ is a minimal sufficient statistic for the uniform$(0,\theta)$ family of distributions."
Theorem 6.6.5 (Minimal sufficient statistics) Suppose that the family of densities $\{f_0(\boldsymbol{X}),...,f_k(\boldsymbol{X})\}$ all have common support. Then
a. The statistic $$T(\boldsymbol{X}) = \bigg( \frac{f_1(\boldsymbol{X})}{f_0(\boldsymbol{X})}, \frac{f_2(\boldsymbol{X})}{f_0(\boldsymbol{X})}, ..., \frac{f_k(\boldsymbol{X})}{f_0(\boldsymbol{X})} \bigg)$$
is minimal sufficient for the family $\{f_0(\boldsymbol{X}),...,f_k(\boldsymbol{X})\}$.
b. If $\mathscr{F}$ is a family of densities with common support, and
(i) $f_i(\boldsymbol{x}) \in \mathscr{F}$, $i=0,1,...,k,$
(ii) $T(\boldsymbol{x})$ is sufficient for $\mathscr{F}$,
then $T(\boldsymbol{x})$ is minimal sufficient for $\mathscr{F}.$
I know how to prove that this statistic is a minimal sufficient statistic for this family of distributions, but I don't know how to prove it this way. Any help would be much appreciated, because I'm getting nowhere with this. Each member of this family of distributions has a different support (namely $(0,\theta)$), so I don't see how this theorem could be applied. But I found a set of notes (https://www.stat.colostate.edu/~riczw/teach/STAT730_S15/Lecture/ST730note.pdf) with the same problem (but no solution), so I'm thinking perhaps this is indeed doable and is not an error in the statement of the problem.