Let $X_1, \ldots, X_n$ be independent random variables with uniform distribution on $[0, \theta] \text{ }, \theta \in [0, \infty)$.
I am to prove that $\max_{i \in \{1, \ldots, n \}}(X_i)$ is a minimal sufficient statistics.
It's easy to show that it is a sufficient statistics (using factorization theorem).
I have problems with showing that it is moreover a minimal statistics.
How can I show that if the ratio
$$\frac{f(x|\theta)}{f(y|\theta)}$$
does not depend on $\theta$ then $\max_{i \in \{1, \ldots, n \}}(X_i) = \max_{i \in \{1, \ldots, n \}}(Y_i)$?