Doubly Robust Property in Causal Inference

• Doubly robust property is defined as an estimator remaining consistent if either the outcome or the missing-data model is correctly specified.
• Its asymptotic error structure factorizes into the product of two estimation errors, isolating bias and highlighting sensitivity to joint misspecification.
• Modern implementations leverage machine learning and robust correction strategies, such as adaptive clipping, to mitigate double fragility and enhance performance.

The doubly robust (DR) property is a foundational concept in semiparametric estimation and causal inference, providing estimators that remain consistent if at least one of two working models—often a model for the outcome and a model for a missingness or treatment assignment process—is correctly specified. This property underpins a broad range of estimators in settings with confounding, missing data, censoring, and high-dimensional nuisance structures, and has far-reaching theoretical and practical implications.

1. Formal Definition and Mechanistic Foundation

A doubly robust estimator is constructed so that its expectation (or, in estimating-function terminology, its unbiasedness) holds as long as either of two nuisance functions is correctly specified. In the prototypical missing data setup, observing data $D_i = (X_i, R_i, R_i Y_i)$, with $R$ a MAR indicator and covariates $X$, the target is the full-data mean $\theta_0 = E[Y]$. With regression function $\mu(x) = E[Y | X = x, R = 1]$ and missingness probability $\pi(x) = P(R = 1 | X = x)$, the canonical doubly robust estimator is

$$\widehat\theta_{\mathrm{DR}} = n^{-1} \sum_{i=1}^n \Bigl[ \widehat\mu(X_i) + \frac{R_i}{\widehat\pi(X_i)} \bigl(Y_i - \widehat\mu(X_i)\bigr) \Bigr]$$

This estimator is consistent for $\theta_0$ if either $\widehat\mu \to \mu$ (with arbitrary $\widehat\pi$) or $\widehat\pi \to \pi$ (with arbitrary $\widehat\mu$), a fact which generalizes to many settings (e.g., average treatment effects, quantile estimation, survival analysis), provided underlying identification assumptions such as MAR, no unmeasured confounding, and positivity hold (Testa et al., 26 Sep 2025).

2. Structure of Asymptotic Error and Double Robustness

A key aspect of DR estimators is that their remaining first-order bias, after Taylor or von Mises expansion, factorizes into a product of the estimation errors of the two involved nuisance functions. For the estimator above, this error can be written as

$$\eta((\widehat\mu,\widehat\pi), (\bar\mu,\bar\pi)) = E\big[(\widehat\mu - \bar\mu)\, (1 - \bar\pi/\widehat\pi)\big]$$

where $(\bar\mu, \bar\pi)$ are the probability limits. The Cauchy–Schwarz inequality gives a dominant bias of $O(\epsilon_g \epsilon_\pi)$ for errors $\epsilon_g = \|\widehat\mu - \bar\mu\|$ and $\epsilon_\pi = \|\widehat\pi - \bar\pi\|$. Thus, if either error vanishes, so does the bias, ensuring DR consistency. However, joint misspecification or slow estimation can render this term non-negligible and potentially large, a phenomenon known as "double fragility" (Testa et al., 26 Sep 2025).

3. Double Fragility, Regularity, and Robustification

While the double robustness property guarantees consistency under partial model correctness, in practice, both nuisance models may be misspecified, rendering the product term non-vanishing and potentially exposing the estimator to bias explosions. Empirical results confirm that, in such double-misspecified regimes, the estimator can perform worse than either of the single-robust plug-in alternatives (outcome regression or inverse-probability weighting) (Testa et al., 26 Sep 2025).

To guard against uncontrolled bias under joint misspecification, robustification strategies—such as adaptive correction clipping (ACC)—ensure that the estimator's error is bounded by a convex combination of the two single-robust estimators' errors:

$$|\widehat\theta_{\mathrm{ACC}} - \theta_0| \leq \lambda \epsilon_g + (1-\lambda) \epsilon_\pi \leq \max(\epsilon_g, \epsilon_\pi)$$

for some data-dependent $\lambda \in [0,1]$. This approach, and variants motivated by similar concerns, replaces the product-structure bias with a sum-structure bound, ensuring no catastrophic failure (Testa et al., 26 Sep 2025). Similar robustification via selective machine learning and ensemble Bayesian synthesis has been advanced in recent literature (Babasaki et al., 10 Sep 2024, Cui et al., 2019).

4. Extension to Survival, Censoring, and Complex Outcomes

The doubly robust framework generalizes to right-censored data, survival, and longitudinal process models by constructing pseudo-outcomes (e.g., via unbiased transformation under coarsening-at-random and positivity) whose conditional bias, again, factorizes into products of estimation errors for regression and censoring nuisance functions. For example, in nonparametric regression with censored data, doubly robust censoring-unbiased transformations achieve DR consistency and oracle efficiency provided that the estimation errors for the censoring hazard and outcome regression are such that their product rate dominates the nonparametric regression rate (Sandqvist, 7 Nov 2024). These ideas have further been instantiated for survival curves, quantiles, empirical kernel statistics, and even for semiparametric estimation in partial interference models (Díaz, 2017, Molina et al., 2017, Fawkes et al., 2022, Liu et al., 2018).

5. Modern Statistical and Machine-Learning Implementations

The double robustness property underpins contemporary methods in double machine learning, targeted regularization, and high-dimensional settings. In double machine learning for semiparametric regression, cross-fitting and orthogonalization produce DR tests and estimators with valid inference as long as at least one nuisance regression (e.g., propensity or outcome) is consistently estimated at sufficient rate; typically, the root-n consistency for the estimator requires the product of convergence rates of both nuisance fits to be $o(n^{-1/2})$ ("rate double robustness") (Dukes et al., 2021, Smucler et al., 2019). Regularization via $\ell_1$ penalties and model selection via pseudo-risk minimization adapted to the DR structure have been shown to deliver near-oracle performance and robust bias control across large ML model libraries (Cui et al., 2019, Smucler et al., 2019).

In Bayesian contexts, DR-type properties are attained via posterior predictive approaches with importance weighting, as standard fully Bayesian approaches with independent priors on nuisance models will generally fail to provide true DR unless the weighting corrects for treatment assignment (Saarela et al., 2017). Bayesian DR ensemble methods further extend consistency guarantees to the case where neither propensity nor outcome model is exactly correct, so long as the truth lies in the convex hull of candidate models (Babasaki et al., 10 Sep 2024).

6. Geometric and Semiparametric Theory Perspectives

Recent developments have provided a geometric and semiparametric theoretical foundation for the doubly robust property. The DR property is characterized as the requirement that an estimating function remains mean-zero if either nuisance is fixed at its truth, irrespective of the other ("global orthogonality"). This requirement is closely related—but not identical—to the influence curve’s (IC's) local orthogonality to the nuisance tangent space. Sufficient conditions for a DR estimator to exist include convexity (m-flatness) of contours in the semiparametric model for each nuisance. If such convexity holds, every efficient IC is automatically DR; otherwise, explicit global orthogonality conditions are needed (Ying, 22 Apr 2024). Information geometry concepts such as e-parallel and m-parallel transport provide further insight into when the DR property is ensured under various model parameterizations.

7. Practical Implications, Diagnostics, and Limitations

The practical advantages of DR estimators are substantial: they offer valid inference under partial knowledge, achieve semiparametric efficiency if both models are correct, and justify the use of flexible, data-adaptive nonparametric or machine-learning fits for nuisance estimation. However, their use demands careful diagnostics for model misspecification and error monitoring; when both nuisance models are poor, mitigation strategies such as correction clipping, robust loss functions, and ensemble averaging should be considered (Testa et al., 26 Sep 2025, Cui et al., 2019).

DR estimators are widely used in observational causal inference, policy evaluation, survival analysis, and high-dimensional statistics, via forms including augmented inverse probability weighting (AIPW), targeted maximum likelihood estimation (TMLE), DR kernel mean and distributional embedding tests, and doubly robust M-estimators for means, quantiles, and more general functionals. The ongoing research agenda emphasizes robustness to joint misspecification, product-structure bias control, post-selection inference, and generalization to complex dependency structures.

---

References

• "Rescuing double robustness: safe estimation under complete misspecification" (Testa et al., 26 Sep 2025)
• "Doubly robust inference with censoring unbiased transformations" (Sandqvist, 7 Nov 2024)
• "On Doubly Robust Inference for Double Machine Learning in Semiparametric Regression" (Dukes et al., 2021)
• "Doubly robust estimation for conditional treatment effect: a study on asymptotics" (Ye et al., 2020)
• "Doubly robust kernel statistics for testing distributional treatment effects" (Fawkes et al., 2022)
• "Statistical Inference for Data-adaptive Doubly Robust Estimators with Survival Outcomes" (Díaz, 2017)
• "Selective machine learning of doubly robust functionals" (Cui et al., 2019)
• "A Geometric Perspective on Double Robustness by Semiparametric Theory and Information Geometry" (Ying, 22 Apr 2024)
• "Ensemble Doubly Robust Bayesian Inference via Regression Synthesis" (Babasaki et al., 10 Sep 2024)