 Doubly robust estimation

Doubly robust estimation combines two nuisance functions so that either one can be correctly estimated while the other is misspecified. For example, let $A\in \{0,1\}$ indicate treatment, $X$ covariates, $m_a(X)=\mathbb{E}[Y\mid A=a,X]$, and $e_a(X)=\mathbb{P}(A=a\mid X)$. Under conditional exchangeability, positivity and the usual consistency of observed and potential outcomes, ${\mu }_a=\mathbb{E}[m_a(X)]$ is the mean potential outcome under treatment $a$.

For candidate functions ${\tilde{m}}_a,{\tilde{e}}_a$ with denominators bounded away from zero, the augmented signal isConditioning on $X$ shows thatThe bias vanishes if either nuisance function is correct. This algebra explains Double robustness and the second-order product bias. It does not protect against both models being wrong, failed Causal identification, or nearly zero denominators. cross-fitting helps when flexible functions are estimated from the same sample; valid confidence intervals require further rate and moment assumptions. The average treatment effect is ${\mu }_1-{\mu }_0$.^[ref]