The best unbiased estimate of a squared probability

Question

You have independent flips X1, \dots, Xn from a Bernoulli(p) and want the best unbiased estimate of 	heta = p^2 (for instance, the chance the next two independent trades both win). Starting from the crude estimator X1 X2, build the minimum-variance unbiased estimator by conditioning on the sufficient total T = \sumi Xi.

Accepted Answer

A crude but honest start. Let W = X1 X2, the product of the first two flips, which is 1 only if both are successes. Since the flips are independent, \mathbb{E}[W] = \mathbb{E}[X1]\mathbb{E}[X2] = p \cdot p = p^2 = 	heta, so W is unbiased. But it uses only two of the n flips. Polish by conditioning (Rao-Blackwell). Average W over all data sharing the same total T = \sumi Xi; the result is unbiased with variance no larger. So \hat	heta = \mathbb{E}[X1 X2 \mid T] = P(X1 = 1, X2 = 1 \mid T). Do the conditional. Given that t of the n flips are successes, every arrangement of which flips succeede…

The best unbiased estimate of a squared probability

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