For what $r,s$ exist unbiased estimation of $f(p) = p^{r}(1 - p)^{s}$ for binomial distribution?

Question

We have sample $x_1, ..., x_n$ generated by independent binomial random variables $\xi_1, ..., \xi_n$.

We know parameter $k$ but don't know probability $p$.

k is number of tests: $\xi_i \sim Binomial(k,p)$

The task is to find numbers $r,s$, that there is exist unbiased estimation for $$f(p) = p^{r}(1 - p)^{s}$$

The problem is that I don't understand the general approach how to test existence of unbiased estimation.

Could you help me please?

Answer 1

In this case, it is equivalent to think that you have an $iid$ sample of $m=nk$ Bernoulli random variables with probability of success being $p$. An unbiased estimator exists when $r+s\leq m-1$. Check out Chapter 2 of Statistical Implications of Turing's Formula (by Zhang), John Wiley & Sons 2017.