Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 6H (Statistics)

A sufficient statistic $T$ is one for which the conditional distribution of the full sample $(X_1,\dots ,X_n)$ given $T$ does not depend on $p$. TakeFor a binary sample $x$ with ${\sum }_i{x}_i=t$, its likelihood function iswhich depends on the data only through $t$. By the Fisher-Neyman factorization theorem, $T$ is sufficient. Equivalently, conditionally on $T=t$, the sample is uniform over the $\left(\genfrac{}{}{0ex}{}{n}{t}\right)$ binary vectors containing $t$ ones, independently of $p$.

The Rao-Blackwell theorem states that if $T$ is sufficient and $U$ is an estimator with finite variance, thenhas the same expectation as $U$ and no larger variance; it preserves unbiasedness. The tower property of conditional expectation givesThe law of total variance givesSufficiency ensures that $U^{\ast }$ is a statistic whose definition does not depend on the unknown parameter. The inequality is strict exactly when the conditional variance is positive with positive probability.

The estimator $U=X_1{X}_2$ is unbiased for $p^2$ because the independent random variables $X_1,X_2$ satisfy $\mathbb{E}[X_1{X}_2]=p^2$. Given $T=t$, all placements of the $t$ successes are equally likely, soThus the Rao-Blackwellized estimator isIt is unbiased by the tower property. Since $n\ge 3$ and $p\in (0,1)$, the event $T=2$ has positive probability, and conditionally on it $X_1{X}_2$ takes both zero and one with positive probability. Hence $\mathbb{E}[\mathrm{Var}(X_1{X}_2\mid T)]>0$, so the new estimator has strictly smaller variance.