Doubly robust estimation and cross-fitting

Let mₐ(L) be the true observed outcome mean in arm a and let e(L) be the true propensity. For candidate functions g₁,g₀ and q, start with the predicted mean contrast g₁ minus g₀. Add the treated residual divided by q and subtract the control residual divided by one minus q. Averaging this score gives the augmented estimator. The candidates must be measurable baseline functions, and q must stay strictly between zero and one where evaluated; otherwise the displayed expression is undefined.

If q equals e, conditional expectation cancels the assignment probabilities, so the residual terms correct both prediction errors exactly. If both g functions equal their true m functions, each arm’s residual has conditional mean zero, leaving the correct predicted contrast even when q is misspecified. These are statements for a fixed candidate score at the population level. Estimated functions introduce dependence on training data, so turning the identity into a sample consistency theorem requires convergence and an appropriate treatment of that dependence.

Subtract the true conditional effect from the conditional expectation of the score. The remaining bias is the propensity error times a weighted combination of outcome-regression errors. If q is bounded away from zero and one, Cauchy–Schwarz bounds its population magnitude by a constant times the propensity L² error multiplied by the sum of the two outcome L² errors. This is the source of first-order insensitivity near the true nuisance functions and the reason good estimation in one component can compensate for slower estimation in the other.

For an ordinary asymptotically linear estimator, a useful sufficient requirement is that this product be smaller in probability than n to the negative one half. If both nuisance errors are smaller than n to the negative one quarter, that requirement follows, but that symmetric rate is not necessary. Mere double-robust consistency with one correct limit does not automatically yield a centered normal interval when the other nuisance remains badly misspecified. Distinguish the zero-bias population branches from the conditions needed for a particular inference theorem.

Partition independent observations into a fixed number of folds. For each fold, fit all nuisance functions using observations outside that fold and evaluate the augmented score only on the held-out observations. Aggregate those scores across folds. A unit’s own outcome is therefore not used to train its score functions. This reduces empirical-process restrictions and overfitting interactions in the proof; it does not make folds’ fitted functions jointly independent or create unmeasured confounder information.

One convenient sufficient regime uses iid observations, bounded outcomes and bounded outcome predictions, true and estimated propensities uniformly bounded away from the endpoints, L²-consistent nuisances, the stated product rate, and a positive limiting influence-score variance. With a fixed fold count these conditions support asymptotic normal inference using the score variance. Clustered data, time series, weak overlap, or adaptive reuse of observations require additional analysis. The method improves estimation of an already identified effect; it cannot turn an unidentified observational contrast into a causal one.