We have a sample $X_i$, where $i=1...n$ and $X_i \sim Poi(\lambda)$. The problem is to find a constant so that $\frac{1}{2} s^2 + cM(X)$, where $s$ is the corrigated sample standard deviation and $M(X)$ is the mean of the sample, is an unbiased estimation of $\lambda$.
Because $s^2$ and $M(X)$ are unbiased estimation of $\lambda$ I would say $c = \frac{1}{2}$ is a good answer.
Am I right? Thank you for your answers