= Inverse-probability-weighted estimator of the average treatment effect
{title2=$\widehat{\operatorname{ATE}}_{\rm IPW}$}
For binary treatment, outcomes $Y_i$, and estimated <propensity score> $\widehat e(X_i)$, the inverse-probability-weighted estimator is
$$
\widehat{\operatorname{ATE}}_{\rm IPW}
=\frac1n\sum_{i=1}^n\left\{
\frac{A_iY_i}{\widehat e(X_i)}-
\frac{(1-A_i)Y_i}{1-\widehat e(X_i)}
\right\}.
$$
It is consistent under <conditional exchangeability>, <consistency of potential outcomes>, <positivity in causal inference>, and a consistent propensity-score estimator.
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