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Give, $Y_1, Y_2, ... Y_n$ are IID random variables, each having a Poisson Distribution with parameter $\lambda$. Is there any unbiased estimator of either (i) $e^{-\lambda}$ or (ii) $e^{-2\lambda}$ whose variance can achieve the Cramer- Rao bound corresponding to them? For $e^{-\lambda}$ , I found the CRLB to be $\dfrac{\lambda}{n}e^{-2\lambda}$ and for $e^{-2\lambda}$ it was $\dfrac{4\lambda}{n}e^{-4\lambda}$.

Could anyone lead me further with this?

Umbrage
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    As $\Pr{Y_1 = 0} = e^{-\lambda}$, you may try to start with the unbiased estimator $\displaystyle \frac {1} {n} \sum_{i=1}^n \mathbf{1}_{Y_i = 0}$, and then apply the Rao-Blackwell + Lehmann–Scheffé theorem – BGM Feb 17 '19 at 02:46
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    https://math.stackexchange.com/q/2690233/ – NCh Feb 17 '19 at 07:09

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