The variance floor for estimating an arrival rate

Question

Requests arrive at a server following a Poisson distribution with unknown rate $\lambda$ per minute. You observe counts $X_1, \dots, X_n$ over $n$ separate minutes.

Find the Cramer-Rao lower bound on the variance of any unbiased estimator of $\lambda$ , and determine whether the sample mean $\bar X$ attains it.

Accepted Answer

What the bound says. Under standard 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 carried by a single observation. An unbiased estimator that reaches this floor is called efficient. Fisher information for one Poisson count. The log-probability of a single count is \log f(x; \lambda) = -\lambda + x\log\lambda - \log(x!). The score (its derivative in \lambda) is s(x;\lambda) = \frac{\partial}{\partial\lambda}\log f = -1 + \frac{x}{\lambda} = \frac{x - \lambda}{\lambda}. Fisher in…

The variance floor for estimating an arrival rate

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