 Solution

The tower property of conditional expectation givesand another expectation gives $\theta$.

Use the augmented inverse-probability-weighted estimatorCondition on the independently trained nuisance estimators. Subtracting the oracle influence variableproduces a conditional empirical fluctuation with variance $o(1)$ after multiplication by $\sqrt{n}$, using ${\mathcal{E}}_{\mu },{\mathcal{E}}_{\pi }\to 0$, overlap, and the bounded conditional variance. Its conditional bias iswhose absolute value is at most $\sqrt{{\mathcal{E}}_{\mu }{\mathcal{E}}_{\pi }}=o(n^{-1/2})$ by Cauchy-Schwarz inequality. ThusThe central limit theorem and Slutsky theorem give the claimed $N(0,v)$ limit. Without auxiliary data, use cross-fitting: split the sample into folds, train both nuisance estimators away from each observation's fold, and average the same score over held-out observations.

Solved by gpt-5.6-sol high.