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Let $x_1 ... x_n$ be $Pois(\lambda)$, $\lambda>0$ and $n \geq 2$. Use $W(X)=I(X_1=0)$ to estimate $g(\lambda)=e^{-\lambda}$. Additionally, $T(X)= \sum_{i=1}^n X_i$ is a sufficient statistic for $\lambda$.

I have already proved that $\phi(T(X))=(1-\frac{1}{n})^{\sum_{i=1}^n X_i}$ is an unbiased estimator of $e^{-\lambda}$.

I want to prove if $\phi(T(X))$ attains the Cramer-Rao lower bound, but i get lost trying to calculate it by applying the definition.

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First, let's calculate the variance of $\phi(T(X))$:

$$\begin{align} \mathbb E[\phi(T(X))^2]&=\sum_{k=1}^\infty \frac{e^{-n\lambda}(n\lambda)^k}{k!}\cdot\left(1-\frac{1}{n}\right)^{2k}\\ &=e^{-n\lambda}\sum_{k=0}^\infty\frac{(n\lambda)^k}{k!}\left(1-\frac 2n+\frac 1{n^2}\right)^k\\ &=e^{-n\lambda}\sum_{k=0}^\infty\frac{1}{k!}\left(n\lambda-2\lambda+\frac{\lambda}{n}\right)^k\\ &=e^{-n\lambda}\cdot e^{n\lambda-2\lambda+\frac\lambda n}\\ &=e^{\lambda\left(\frac{1}{n}-2\right)} \end{align}$$

And, using the fact that $\phi(T(X))$ is unbiased:

$$\begin{align} \text{Var}(\phi(T(X)))&=\mathbb E[\phi(T(X))^2]-\mathbb E[\phi(T(X))]^2\\ &=e^{\lambda\left(\frac{1}{n}-2\right)}-e^{-2\lambda}\\ &=e^{-2\lambda}\cdot\left(e^{\lambda/n}-1\right) \end{align}$$

Now, we have to compare this to the Cramér-Rao bound. The likelihood function is given by:

$$\begin{align} \ell(\lambda)&=\sum_{i=1}^n\log\left(\frac{e^{-\lambda}\lambda^{X_i}}{X_i!}\right)=-n\lambda + \log(\lambda)T(X)-\sum_{i=1}^n\log(X_i!) \end{align}$$

Then, we can get the Fisher information:

$$\begin{align} \mathcal I_F&=-\mathbb E\left[\frac{\partial^2\ell(\lambda)}{\partial\lambda^2}\right]=\mathbb E\left[\frac{T(X)}{\lambda^2}\right]=\frac n\lambda \end{align}$$

Finally, the Cramér-Rao bound for $g(\lambda)$ is:

$$\frac{g'(\lambda)^2}{\mathcal I_F}=\frac{e^{-2\lambda}}{n/\lambda}=\frac{\lambda}{n}e^{-2\lambda}$$

Which is different from the variance we previously found, so the estimator you found does not attain the Cramér-Rao bound. You can also prove, using Taylor expansion, that your estimator is asymptoticaly efficient, that is, the variance of the estimator is asymptotically equal to the Cramér-Rao bound.

Hope it was helpful!

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