Information for a Poisson mean

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Let X_1,\ldots,X_n be a random sample from the Poisson distribution with mean \lambda>0.

1. Write the score and the expected information for \lambda.

2. Show that \overline X attains the Cramér–Rao bound for \lambda.

Worked answer

The log-likelihood is \ell(\lambda)=-n\lambda+(\sum_j X_j)\log\lambda, up to a term free of \lambda. The score is

S(\lambda)=\frac{1}{\lambda}\sum_{j=1}^n X_j-n=\frac{n}{\lambda}(\overline X-\lambda).

Then \mathbb{E} S(\lambda)=0, and

\mathcal{I}(\lambda)=\operatorname{Var}\{S(\lambda)\}=\frac{n^2}{\lambda^2}\operatorname{Var}(\overline X)=\frac{n^2}{\lambda^2}\cdot\frac{\lambda}{n}=\frac{n}{\lambda}.

The Cramér–Rao bound for an unbiased estimator of \lambda is 1/\mathcal{I}(\lambda)=\lambda/n. The sample mean is unbiased and \operatorname{Var}(\overline X)=\lambda/n, so it attains the bound. The same conclusion is the factorisation S(\lambda)=(n/\lambda)(\overline X-\lambda).