Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 4

For binary $A$, conditional covariance satisfiesBy conditional exchangeability and consistency of potential outcomes, the difference in braces is $\mathbb{E}[Y(1)-Y(0)\mid X]$. Consequentlyso the required weight isThis overlap weight emphasizes covariate strata with treatment probabilities near one half and downweights strata near a violation of positivity in causal inference.

Write $p_x=\mathbb{P}(X=x)$, ${\pi }_x=\mathbb{E}(A\mid X=x)$, and ${\mu }_x=\mathbb{E}(Y\mid X=x)$. The functional isThe first term has influence function $AY-\mathbb{E}(AY)$. Applying the product rule to the second term, including perturbations of $p_x$, ${\pi }_x$, and ${\mu }_x$, givesSubtracting and using $\beta =\mathbb{E}(AY)-\mathbb{E}[\pi \mu ]$ yieldsIts expectation is zero because the first product has expectation $\mathbb{E}\{\mathrm{Cov}(A,Y\mid X)\}=\beta$.

Condition on the independent external sample, and abbreviate ${\delta }_{\pi }={\hat{\pi }}_m-\pi$ and ${\delta }_{\mu }={\hat{\mu }}_m-\mu$. Expanding the plug-in estimator around the true nuisance functions giveswhere $\sqrt{n}{R}_{n,m}=o_p(1)$ because the two nuisance mean squared errors vanish, the evaluation sample is independent of the nuisance fits, $A$ is bounded, and $Y$ has bounded support. The linear term is the empirical average of the influence function, so the central limit theorem givesThe remaining bias obeys the Cauchy-Schwarz inequalityThus the claimed conclusion follows under the standard product-rate conditionequivalently $\sqrt{n}\sqrt{\mathrm{MSE}({\hat{\pi }}_m)operatornameMSE({\hat{\mu }}_m)}\to 0$. Slutsky theorem then yields

As printed, the paper instead assumes only $\sqrt{n}\mathrm{MSE}({\hat{\pi }}_m)\mathrm{MSE}({\hat{\mu }}_m)\to 0$, which is insufficient with its stated definition of MSE. For example, deterministic nuisance errors of size $n^{-3/16}$ have both MSEs equal to $n^{-3/8}$ and satisfy the printed condition, while the scaled product bias is $\sqrt{n}{n}^{-3/8}=n^{1/8}$. The result therefore requires the stronger condition above, or “MSE” in the printed rate must be read as root mean squared error.