For what values of $k \in \mathbb{Z}\setminus\{0\}$ does there exist an unbiased estimator of $e^{k \lambda}$?
Here are some unbiased estimators of $e^{a\lambda}$, for every nonnegative $a$, based on any i.i.d. sample $(X_1,\ldots,X_n)$ of Poisson distribution with parameter $\lambda$:
- Using $X_1$ only, $$(1+a)^{X_1}$$
- Using $(X_1,\ldots,X_n)$, $$\left(1+\frac an\right)^{X_1+X_2+\cdots+X_n}$$
- If asymmetry is allowed, $$\prod_{i=1}^na_i^{X_i}$$ for every real numbers $(a_i)$ such that $$\sum_{i=1}^na_i=n+a$$
And so on.