= Augmented inverse-probability-weighted estimator
An <augmented inverse-probability-weighted estimator> averages a regression prediction plus an inverse-probability-weighted <regression residual>[residual]. In the treatment setting of <doubly robust estimation>, let $\widehat m_a(X_i)$ estimate the conditional outcome <mean> and $\widehat e_a(X_i)$ the <probability> of receiving treatment $a$. Then
$$
\widehat\mu_a=\frac1n\sum_{i=1}^n\left[\widehat m_a(X_i)+\frac{\mathbf1_{\{A_i=a\}}}{\widehat e_a(X_i)}\{Y_i-\widehat m_a(X_i)\}\right],\qquad \widehat\tau=\widehat\mu_1-\widehat\mu_0.
$$
The first term predicts outcomes for everyone; the weighted <regression residual>[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 <regression residual>[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 <probability>[probabilities] must remain positive on the population being studied, and small fitted <probability>[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 interval>[confidence intervals].
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