Unbiased estimator and sufficient statistic from discrete uniform distribution $z_1,...z_n$ is a sample from a discrete $\{1,...,N\}$ uniform distribution.
I have two questions:


*

*I want to find an unbiased estimator for N, with the help of $z_1$

*I want to find a 1 dimensional sufficient statistic for N.
For 1, I have no idea.
For 2, I would say it is the biggest element of the sample.
 A: *

*To find an unbiased estimator for $N$ using $z_1$, start from here: if $z_1 \sim \textrm{Unif}(1, N)$, then we want to find some function of $z_1$ such that $E(f(z_1)) = N$ (that's just what it means to be unbiased). Since $E(z_1) = \frac1N \sum_{i=1}^N i = \frac{N(N + 1)}{2N} = (N+1)/2$, it's hopefully pretty easy to come up with such a function...

*You're on the right track. To prove it's sufficient you can use the Fisher-Neyman Factorization Theorem: a statistic $T(z_1, \dots, z_n)$ is sufficient for $N$ if the joint CDF $f_N(z_1, \dots, z_n)$ can be expressed as a product of two terms, one of which does not depend on $N$ (that is, it depends only on the $z_i$) and the other of which depends only on $N$ and $T$ (so not on the individual $z_1$, only on in this case their maximum). So you need only think of a way to express the joint CDF of the uniform distribution in this form, and then you're done.
Note: I suggest you don't read the rest of that Wikipedia page (only the section I linked to), as it spoils the answer to this problem.
