Cramer-Rao bound for a Poisson rate

Question

Let X1, \dots, Xn be i.i.d. Poisson(\lambda) (for example, counts of trades arriving each second). State the Cramer-Rao lower bound, compute the Fisher information for \lambda, and determine whether \hat{\lambda} = \bar{X} is efficient.

Accepted Answer

The bound. Under regularity conditions, any unbiased estimator T of \lambda satisfies \operatorname{Var}(T) \ge \frac{1}{n\,I(\lambda)}, where I(\lambda) is the Fisher information of a single observation. An unbiased estimator meeting the bound is efficient. Fisher information. The Poisson log-density of one observation is \log f(x; \lambda) = -\lambda + x\log\lambda - \log(x!), so the score is s(x; \lambda) = \frac{\partial}{\partial \lambda}\log f = -1 + \frac{x}{\lambda} = \frac{x - \lambda}{\lambda}. (Sanity check: E[s] = 	frac{E[X] - \lambda}{\lambda} = 0, as a score must be.) The infor…

Cramer-Rao bound for a Poisson rate

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