The best unbiased estimate of a squared Poisson rate

Question

You have independent counts X1, \dots, Xn from a Poisson(\lambda) and want the best unbiased estimate of 	heta = \lambda^2. Starting from the crude estimator X1 X2, build the minimum-variance unbiased estimator by conditioning on the sufficient total T = \sumi Xi.

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Why not just square the mean. The tempting \bar X^2 is biased: \mathbb{E}[\bar X^2] = \lambda^2 + \lambda/n \lambda^2, so it overshoots. We build an exactly unbiased estimator instead. A crude but honest start. Let W = X1 X2, the product of two independent counts. By independence, \mathbb{E}[W] = \mathbb{E}[X1]\mathbb{E}[X2] = \lambda \cdot \lambda = \lambda^2 = 	heta, so W is unbiased. Polish by conditioning (Rao-Blackwell). Average W over all data with the same total T = \sumi Xi: \hat	heta = \mathbb{E}[X1 X2 \mid T]. Do the conditional. Given T = t, the counts split like a multinomial th…

The best unbiased estimate of a squared Poisson rate

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