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Let $X_1, \cdots, X_n$ be iid with $P(X_i=j)=1/N$, for $j=1, \cdots, N$, where $N$ is an unknown positive integer.

I would guess that the order statistics $X_{(n)}$ is the complete sufficient statistic of $N$, but I'm having trouble in proving this.

Also, I understand that best unbiased estimator of $N$ could be in the form of $2E(X_1|X_{(n)})-1$, but I'm having trouble to go further.

Any help would be appreciated!

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