The variance floor for the chance of an idle minute

Question

Arrivals in a minute are Poisson with unknown rate $\lambda$ , and you observe $n$ minutes. Instead of the rate, you care about $g(\lambda) = e^{-\lambda}$ , the probability that a given minute is idle (zero arrivals).

State the Cramer-Rao lower bound for an unbiased estimator of $g(\lambda) = e^{-\lambda}$ .

Accepted Answer

The bound for a transformed parameter. When you estimate a smooth function g(\lambda) rather than \lambda itself, the Cramer-Rao lower bound picks up the squared derivative of g: \operatorname{Var}(T) \ge \frac{[g'(\lambda)]^2}{n\,I(\lambda)}. The factor [g'(\lambda)]^2 rescales the floor by how fast the target changes with the parameter. Plug in the Poisson pieces. For a single Poisson count the Fisher information is I(\lambda) = \frac{1}{\lambda}. The target and its derivative are g(\lambda) = e^{-\lambda}, \qquad g'(\lambda) = -e^{-\lambda}, \qquad [g'(\lambda)]^2 = e^{-2\lambda}. Assemble…

The variance floor for the chance of an idle minute

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