Note on the Unbiased Estimation of a Function of the Parameter of the Geometric Distribution

Kolmogorov studied the problem of whether a function of the parameter p of the Bernoulli distribution Bernoulli[p] has an unbiased estimator based on a sample X-1, X-2, ... , X-n of size n and proved that exactly the polynomial functions of degree at most n can be estimated. For the geometric distribution Geometric[p], we prove that exactly the functions that are analytic at p = 1 have unbiased estimators and present the best estimators.