Deep Neural Networks for Doubly Robust Estimation with Nonprobability Survey Samples

Abstract: Integrating probability and nonprobability survey samples is an important problem in modern survey sampling. Nonprobability samples often contain rich outcome information but may lack population representativeness, whereas probability samples provide design-based auxiliary information but may not contain the study variable. We propose a deep neural network (DNN)-assisted doubly robust framework for estimating the finite population mean from these two data sources. The proposed method models the logit sampling score for the nonprobability sample as an unknown nonparametric function and estimates it by maximizing a pseudo-likelihood that combines information from the nonprobability sample and a reference probability sample. The DNN parameters are optimized using the ADAM algorithm. The resulting DNN-estimated sampling scores are incorporated into a DNN-assisted inverse-probability weighted estimator and a deep doubly robust estimator. We establish consistency and convergence rates under regularity conditions and evaluate the finite-sample performance of the proposed estimators through simulation studies and an empirical application using Pew Research Center and Behavioral Risk Factor Surveillance System data. The results suggest that the proposed estimators can improve robustness to parametric propensity-score misspecification, especially when the true selection mechanism is nonlinear.

• The paper introduces a DNN framework that uses nonparametric modeling of the logit sampling score for robust doubly robust estimation.
• It integrates DNN-derived propensity scores in both inverse probability weighting and outcome regression to mitigate bias under nonlinear selection.
• Empirical simulations and real-world data demonstrate near-zero bias and reduced mean squared error compared to traditional methods.

Deep Neural Networks for Doubly Robust Estimation in Nonprobability Survey Samples

Background and Motivation

Nonprobability survey samples, often arising from web panels, sensor data, and satellite information, present significant inferential challenges due to selection bias and lack of population representativeness. Conversely, probability samples offer design-based auxiliary information but may omit key outcome variables. Recent literature divides integration strategies into calibration weighting, mass imputation, and propensity score adjustment. However, existing methodologies relying on parametric model assumptions are susceptible to bias under model misspecification.

Doubly robust (DR) estimators combine outcome regression and propensity score modeling, achieving consistency if either component is correctly specified. Nonetheless, their performance degrades when both models are misspecified, particularly in the presence of nonlinear selection mechanisms. Machine learning methods, and specifically deep neural networks (DNNs), offer flexible, high-capacity alternatives for capturing complex nonlinearities.

Methodological Contribution

The paper introduces a DNN-assisted estimation framework for finite population means in nonprobability surveys (2605.28762). The approach models the logit sampling score function nonparametrically—eschewing linearity—and fits this via maximization of a pseudo-likelihood that leverages both nonprobability and reference probability samples. Optimization is performed with the ADAM algorithm, ensuring scalability and efficient parameter convergence.

DNN-estimated sampling scores are integrated into two estimators:

• DNN-Inverse Probability Weighting (DIPW): Utilizes DNN-derived propensity scores in an IPW estimator.
• Deep Doubly Robust (DDR) Estimator: Combines nonparametric propensity scores with outcome regression, enhancing robustness.

The theoretical results establish consistency and convergence rates for these estimators under composite smoothness and sparsity assumptions, demonstrating that convergence depends on the intrinsic dimensionality rather than the ambient dimension. This mitigates the curse of dimensionality, enabling efficient inference in high-dimensional settings.

Simulation and Empirical Results

Extensive simulations use a population model with nonlinear selection mechanisms—propensity scores involve interactions and non-linear terms omitted in standard parametric specifications. The simulations examine scenarios with misspecified propensity models, misspecified outcome regression, and both misspecified ("TF" and "FF" scenarios).

Key empirical findings:

• DDR estimator outperforms conventional DR, REG, and IPW estimators when the selection mechanism is nonlinear and parametric models are misspecified. In these settings, DIPW and DDR estimators exhibit near-zero bias and substantially reduced mean squared error (MSE).
• Conventional parametric estimators demonstrate marked bias and inflated MSE under misspecification, confirming the vulnerability of linear or logistic regression approaches in complex survey settings.
• Applied to Pew Research Center and Behavioral Risk Factor Surveillance System data, DNN-based estimators produce population mean estimates aligned with adjusted probability-sample estimates, with departures observed on specific outcomes sensitive to selection bias.

Theoretical Implications and Extensions

Theoretical results demonstrate that, under regularity and composite smoothness assumptions, the DNN-based estimators achieve $O_p(\gamma_n\log^2 n)$ convergence rates, where $\gamma_n$ depends on smoothness and intrinsic dimensionality. These rates are competitive with minimax optimal rates for nonparametric regression with DNNs, confirming the adaptability of the proposed approach.

The method reduces sensitivity to propensity score specification, offering increased protection against bias in settings with unknown or nonlinear selection mechanisms. However, identification assumptions remain crucial: rich auxiliary covariates and positivity of inclusion probabilities are required for estimator validity.

The paper advocates extension to fully nonparametric regression modeling via DNNs in both outcome and propensity functions, anticipating further performance gains in highly complex survey scenarios. Such developments would introduce additional computational and theoretical challenges, but are a promising direction for advancing robust survey methodology.

Practical Implications and Future Directions

Practically, the methodology is directly applicable to official statistics, epidemiology, and social science research where heterogeneous survey data must be integrated for population inference. The DNN-assisted DR estimator provides a flexible tool for practitioners, enabling valid inference without stringent parametric assumptions or prior knowledge of population-level auxiliary distributions.

Future work should address:

• Joint nonparametric modeling of outcome and propensity functions using DNNs, with theoretical extension to double deep robustness.
• Adaptation of the framework to settings with incomplete auxiliary information or partial overlap between probability and nonprobability samples.
• Development of diagnostic tools for assessing model fit and identifying departures from identification assumptions in applied settings.

Conclusion

This work advances survey inference by integrating deep learning into doubly robust estimation, achieving reliable and efficient population mean estimation with heterogeneous and potentially biased data sources (2605.28762). The empirical and theoretical findings highlight the benefit of DNNs in alleviating model misspecification, particularly in the presence of nonlinear selection mechanisms. The proposed methodology sets the stage for future research in fully nonparametric survey inference and practical data integration, with broad implications for statistical methodology and the development of robust AI-driven approaches in survey analytics.