Rao-Blackwell in action on a Poisson probability

Question

Let X1, \dots, Xn \overset{iid}{\sim} 	ext{Poisson}(\lambda), and suppose you want to estimate 	au = P(X = 0) = e^{-\lambda}. Start from the crude unbiased estimator \delta = \mathbf{1}\{X1 = 0\}, Rao-Blackwellize it on the sufficient statistic T = \sumi Xi, and identify the result.

Accepted Answer

The crude estimator. \delta = \mathbf{1}\{X1 = 0\} is unbiased, since \mathbb{E}[\delta] = P(X1 = 0) = e^{-\lambda} = 	au. But it is terrible: it ignores X2, \dots, Xn and takes only the values 0 or 1. Rao-Blackwell. Conditioning any unbiased estimator on a sufficient statistic T gives \delta^\star = \mathbb{E}[\delta \mid T], which stays unbiased and has variance no larger (by the law of total variance). Here T = \sumi Xi is sufficient for \lambda. Compute the conditional expectation: \delta^\star = \mathbb{E}\big[\mathbf{1}\{X1 = 0\} \mid T = t\big] = P(X1 = 0 \mid T = t). The key distribu…

Rao-Blackwell in action on a Poisson probability

Your answer

More Statistics questions