Doubly robust estimators combine outcome regression and propensity weighting so that consistency survives misspecification of either nuisance model. Their deeper value is Neyman orthogonality: nuisance errors enter the target estimate primarily through a product.
The augmented inverse-probability estimator
Definition. Let \(m_a(x)=\mathbb E(Y\mid A=a,X=x)\) and \(e(x)=P(A=1\mid X=x)\). Define \[ \psi(O;\eta)= m_1(X)-m_0(X) +\frac{A\{Y-m_1(X)\}}{e(X)} -\frac{(1-A)\{Y-m_0(X)\}}{1-e(X)}, \] where \(\eta=(m_0,m_1,e)\). The AIPW estimator is \[ \widehat\tau=P_n\psi(O;\widehat\eta). \]
Double robustness
Theorem. Let \(\bar m_a,\bar e\) be probability limits of nuisance estimators, bounded away from positivity boundaries. Under causal identification, \[ P\psi(O;\bar m,\bar e)=\tau \] if either \(\bar m_a=m_a\) for both arms or \(\bar e=e\).
Proof. Condition on \(X\). The treated residual term has mean \[ \frac{e(X)}{\bar e(X)} \{m_1(X)-\bar m_1(X)\}, \] and the control residual term has mean \[ -\frac{1-e(X)}{1-\bar e(X)} \{m_0(X)-\bar m_0(X)\}. \] Adding \(\bar m_1-\bar m_0\) equals \(m_1-m_0\) if \(\bar m=m\), because both residual means vanish, or if \(\bar e=e\), because each regression error is exactly corrected. Integrate.
Exact remainder identity
Proposition. For generic nuisances \(\bar\eta\), \[ P\psi(O;\bar\eta)-\tau = \mathbb E\left[ \frac{\bar e-e}{\bar e} (\bar m_1-m_1) + \frac{\bar e-e}{1-\bar e} (\bar m_0-m_0) \right]. \]
Proof. Use the conditional expectation calculation from the preceding proof and collect coefficients of each outcome-regression error. No asymptotic approximation is involved.
Corollary. Under strong overlap, \[ |P\psi(\bar\eta)-\tau| \lesssim \|\bar e-e\|_{P,2} \{\|\bar m_1-m_1\|_{P,2}+ \|\bar m_0-m_0\|_{P,2}\}. \]
Proof. Bound inverse propensity factors by overlap constants and apply Cauchy-Schwarz to each product.
Neyman orthogonality
Definition. A score \(\psi(O;\tau,\eta)\) is Neyman-orthogonal at \((\tau_0,\eta_0)\) if the pathwise derivative \[ \left.\frac{\mathrm{d}}{\mathrm{d}t} P\psi(O;\tau_0,\eta_0+t h)\right|_{t=0}=0 \] for every admissible nuisance direction \(h\).
Proposition. The AIPW score is Neyman-orthogonal.
Proof. The exact bias identity is a product of propensity and outcome-regression errors. Along a path \(\eta_t=\eta_0+t h\), it is \(O(t^2)\), so its derivative at zero vanishes.
Cross-fitting
Definition. Partition observations into folds \(I_1,\ldots,I_K\). Fit nuisances \(\widehat\eta^{(-k)}\) without fold \(k\), and set \[ \widehat\tau_{\mathrm{CF}} =\frac1n\sum_{k=1}^K\sum_{i\in I_k} \psi(O_i;\widehat\eta^{(-k)}). \]
Conditional on training folds, evaluation observations are independent of their nuisance fits. This avoids Donsker restrictions on highly adaptive learners and lets empirical-process terms be controlled by conditional variance.
Asymptotic linearity
Theorem. Suppose overlap holds, nuisances are \(L^2(P)\)-consistent, their product error is \(o_P(n^{-1/2})\), and moment conditions hold. Then \[ \sqrt n(\widehat\tau_{\mathrm{CF}}-\tau) = \frac1{\sqrt n}\sum_{i=1}^n\phi_{\mathrm{eff}}(O_i)+o_P(1) \rightsquigarrow N(0,V_{\mathrm{eff}}), \] where \[ \phi_{\mathrm{eff}}(O)=\psi(O;\eta_0)-\tau. \]
Proof outline. Add and subtract \(P_n\psi(\eta_0)\). The population nuisance bias is the second-order product remainder. Cross-fitting and conditional variance make \((P_n-P)\{\psi(\widehat\eta)-\psi(\eta_0)\}=o_P(n^{-1/2})\) under \(L^2\)-consistency. The remaining term is the iid average of the efficient influence function, so the central limit theorem applies.
Rate combinations
The sufficient product condition allows asymmetric nuisance rates. If the propensity is parametric at \(n^{-1/2}\), the outcome regression needs only consistency. If both nuisances converge at \(o_P(n^{-1/4})\), their product is \(o_P(n^{-1/2})\). Exact rates depend on norms, overlap, and whether one nuisance is known.
Targeted minimum-loss estimation
TMLE begins with initial nuisance estimates and updates the outcome regression along a least-favorable fluctuation with clever covariate \[ H(A,X)=\frac{A}{\widehat e(X)}- \frac{1-A}{1-\widehat e(X)}. \] The update solves the empirical efficient-score equation, after which the target is evaluated as a substitution estimator. TMLE and one-step AIPW have the same first-order influence function but differ in finite-sample substitution and range properties.
What double robustness does not mean
It does not mean the estimator is reliable when both models are badly wrong, immune to weak overlap, or free from regularity conditions. Near positivity boundaries, small nuisance errors are amplified. Finite-sample bias can be severe even when one nuisance is asymptotically correct. Diagnostics and sensitivity remain necessary.