I'm trying to find the minimally sufficient statistic where $\{X_i\}_{i=1}^{n}$ are iid from the following family of populations:
$$P=\{U(0,\theta): \theta>0\}$$
I looked at the ratio of the likelihood of two observations in the support of the family of populations:
$$\frac{\frac{1}{\theta}^n\chi_{(x_{(n)},\infty)}(\theta)}{\frac{1}{\theta}^n\chi_{(y_{(n)},\infty)}(\theta)}=\frac{\chi_{(x_{(n)},\infty)}(\theta)}{\chi_{(x_{(n)},\infty)}(\theta)}$$
This will not be a function of $\theta$ iff $X_{(n)}=Y_{(n)}$. One can conclude that $X_{(n)}$ is a minimally sufficient statistic. I am questioning my answer because it is my understanding that the ratio cannot include $\theta$, I have only shown is that it will not depend on $\theta$ if $X_{(n)}=Y_{(n)}$. Is this the same or am I missing anything?
