 Augmented inverse-probability-weighted estimator

An augmented inverse-probability-weighted estimator averages a regression prediction plus an inverse-probability-weighted residual. In the treatment setting of doubly robust estimation, let ${\hat{m}}_a(X_i)$ estimate the conditional outcome mean and ${\hat{e}}_a(X_i)$ the probability of receiving treatment $a$. ThenThe first term predicts outcomes for everyone; the weighted residual corrects prediction errors using observed outcomes from the corresponding treatment group. The conditional-expectation calculation in doubly robust estimation makes the remaining bias a product of outcome-regression and propensity-score errors. An exactly correct outcome regression makes the residual correction mean zero even with an incorrect propensity score; a correct propensity score makes that correction repair the regression bias.

The nuisance estimates are commonly obtained by cross-fitting, with each observation evaluated using functions trained on other observations. Both treatment probabilities must remain positive on the population being studied, and small fitted probabilities can make the variance very large. Double robustness concerns consistency under one correct nuisance model; it does not automatically deliver valid finite-sample or asymptotic confidence intervals.