Find a comlete and sufficient statistics for $U(\theta,\theta)$. I know that for $U(a,b)$ the complete and sufficient statistic is $(X_{(1)},X_{(n)})$ but I'm not sure if for this one is just $X_{(n)}$
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1And what if your sample were $(10, 7, 1, 3, 4)$? Then $X_{(n)} = 4$, but this discards information in the sample that implies we must have $\theta \ge 10$. – heropup May 20 '21 at 19:07

If $X_i\sim U(\theta,\theta)$, then $X_i\sim U(0,\theta)$. – StubbornAtom May 20 '21 at 20:20

So essentially a duplicate of https://math.stackexchange.com/q/699997/321264, https://math.stackexchange.com/q/1973864/321264. – StubbornAtom May 20 '21 at 20:37