The best unbiased estimate of a Bernoulli variance

Question

You have independent flips X1, \dots, Xn from a Bernoulli(p) and want the best unbiased estimate of 	heta = p(1 - p), the variance of a single flip. Starting from the crude estimator X1(1 - 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(1 - X2), which is 1 exactly when the first flip is a success and the second a failure. By independence, \mathbb{E}[W] = p(1 - p) = 	heta, so W is unbiased. It uses just two flips and is a blunt instrument. Polish by conditioning (Rao-Blackwell). Average W over all data with the same total T; the result stays unbiased and cannot have larger variance: \hat	heta = \mathbb{E}[X1(1 - X2) \mid T] = P(X1 = 1, X2 = 0 \mid T). Do the conditional. Given t successes among n flips, the chance that a specific flip is a success and another specific flip is a failure i…

The best unbiased estimate of a Bernoulli variance

Your answer

Discussion

Learn the concepts

Related questions