Consider $X_{1} \dots X_{n}$ are i.r.v with $Exp(\theta)$ ($p_{\theta} = \theta e^{-\theta x}$). We want to find optimal estimator of $\theta$ (i.e. unbiased estimator with minimal variance in unbiased estimator class).
Our teacher tells us that it can be done with using sufficient and completeness statistics.
It's easy to determine that $X_{(1)}$ is satisfy this properties. Now we want to make it unbiased.
We need to find some $f(x)$, such as : $\mathbb{E}(f(X_{(1)}) = \theta$.
That is a problem moment. Actually I'm thinking that there is no such functions.
I've tried to consider some easy examples : $f(x) = x^{m}, f(x) = \log{x}$ and $f(x) = e^{x}$ but it give's me only $\theta^{a}$ ,where $a \ne 1$ or functions like $\frac{\theta^{m}}{\theta^{2} + const}$ .
Any hint's will be good .
