Rao-Blackwell builds an unbiased estimate of a squared rate

Question

Counts X1, \dots, Xn are i.i.d. Poisson with unknown rate \lambda. You want to estimate \lambda^2. A crude unbiased estimator is T = X1 X2, because independence gives \mathbb{E}[X1 X2] = \lambda \cdot \lambda = \lambda^2. Rao-Blackwellize T by conditioning on the sufficient statistic S = \sumi Xi, and give the resulting unbiased estimator of \lambda^2.

Accepted Answer

Setup. The sum S = \sumi Xi is sufficient for \lambda. Rao-Blackwell says T^\star = \mathbb{E}[X1 X2 \mid S] is unbiased for \lambda^2 with variance no larger than the crude product. The key conditional fact. Given S = s, the vector (X1, \dots, Xn) is Multinomial with s total events dropped independently into n equally likely bins, each with probability 1/n. For a multinomial, two distinct bin counts satisfy \mathbb{E}[X1 X2 \mid S = s] = s(s-1)\cdot\frac{1}{n}\cdot\frac{1}{n} = \frac{s(s-1)}{n^2}. So the improved estimator is T^\star = \frac{S(S-1)}{n^2}. Check it is unbiased. The total S is…

Rao-Blackwell builds an unbiased estimate of a squared rate

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