Deep Doubly Robust Estimator

• The paper introduces a DNN-assisted framework that models the nonparametric logit sampling score to estimate finite-population means.
• It integrates rich outcome data from nonprobability surveys with design-based information from probability samples to mitigate nonlinear selection bias.
• The method achieves near-minimax convergence rates, yielding lower bias and mean-squared error compared to traditional IPW and DR estimators.

Searching arXiv for the specified paper and closely related doubly robust deep-estimation work. Use the arXiv search tool now. Integrating probability and nonprobability survey samples has become a central problem in modern survey sampling because the two sources typically provide complementary information: 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. In this setting, the Deep Doubly Robust Estimator denotes a deep neural network (DNN)-assisted doubly robust framework for estimating the finite population mean from these two data sources by modeling the logit sampling score for the nonprobability sample as an unknown nonparametric function and combining that model with a parametric outcome regression (Dai et al., 27 May 2026). The resulting estimator retains the classical doubly robust structure, but replaces a parametric propensity specification with a DNN fit obtained from a pseudo-likelihood based on the nonprobability sample and a reference probability sample (Dai et al., 27 May 2026).

1. Problem setting and estimand

The framework considers a finite population $\mathcal U$ of size $N$, a nonprobability sample $S_A$ of size $n_A$, and a reference probability sample $S_B$. For each unit $i$, the observed covariates are $x_i\in\mathbb R^r$, and the nonprobability-sample inclusion indicator is

$R_i = 1\{i\in S_A\}.$

The participation, or sampling, probability is defined as

$\pi^A(x_i)=P(R_i=1\mid x_i).$

The target quantity is the finite-population mean

$\mu_y = N^{-1}\sum_{i=1}^N y_i.$

In practice, the nonprobability sample contains $N$0, while the reference probability sample contains $N$1, where $N$2 are design-weights (Dai et al., 27 May 2026).

This formulation separates the inferential roles of the two samples. The nonprobability sample contributes outcome information, and the probability sample contributes design-based information used to approximate population-level covariate structure. A plausible implication is that the method is tailored to settings in which direct design-based estimation is infeasible because the study variable is unavailable in the probability sample, but covariate alignment across the two samples is available.

2. Sampling-score modeling with deep neural networks

Rather than posit a linear logistic model, the method specifies

$N$3

where $N$4 is an unknown smooth function on $N$5. The core idea is to approximate $N$6 by a deep neural network $N$7 with parameters $N$8 (Dai et al., 27 May 2026).

A $N$9-layer feedforward network with layer widths $S_A$0 is defined recursively by

$S_A$1

where $S_A$2 is e.g. the ReLU activation, and $S_A$3 are the trainable weights and biases collected into $S_A$4 (Dai et al., 27 May 2026).

The methodological significance of this replacement is explicit: the DNN is used to estimate the sampling score nonparametrically rather than through a parametric logistic link with fixed linear structure. The reported motivation is robustness to parametric propensity-score misspecification, especially when the true selection mechanism is nonlinear (Dai et al., 27 May 2026). This suggests that the deep component is not introduced as a generic predictor, but specifically as a nuisance-function estimator intended to stabilize downstream inverse-probability and doubly robust estimators under nonlinear selection.

3. Pseudo-likelihood construction and optimization

If $S_A$5 were known for all $S_A$6, the population log-likelihood would be

$S_A$7

Because the full population is not observed, the unknown sum over $S_A$8 is replaced by the Horvitz–Thompson estimator, yielding the pseudo-log-likelihood

$S_A$9

The DNN is then fit by

$n_A$0

using the ADAM stochastic-gradient algorithm, together with early stopping and weight/bias regularization (Dai et al., 27 May 2026).

This estimation procedure is specific to survey integration. The pseudo-likelihood combines information from the nonprobability sample and the reference probability sample rather than treating the sampling-score task as an ordinary supervised classification problem. In that sense, the DNN is embedded within a design-based construction. A plausible implication is that the estimator inherits properties from both semiparametric missing-data methods and finite-population survey inference.

4. Estimator definitions: DNN-assisted IPW and deep doubly robust estimation

Once $n_A$1 is obtained, the estimated sampling scores are

$n_A$2

These scores define the DNN-assisted inverse-probability weighted estimator of the finite-population mean: $n_A$3 Each unit in $n_A$4 therefore receives weight $n_A$5 (Dai et al., 27 May 2026).

The deep doubly robust estimator additionally uses a parametric outcome regression $n_A$6. Let $n_A$7 be its least-squares or GLM estimate on $n_A$8. The estimator is

$n_A$9

Its double robustness is stated as follows: if either the sampling-score model $S_B$0 is correctly estimated by the DNN, or the outcome regression $S_B$1 is correctly specified, then $S_B$2 is consistent for $S_B$3 (Dai et al., 27 May 2026).

The structure is a direct extension of classical doubly robust estimation. The innovation is localized in the propensity component: the outcome model remains parametric, while the propensity side is replaced with a DNN-estimated nonparametric logit score. This suggests that the method is best understood as a deep nuisance-model substitution within a conventional augmentation architecture, rather than as an entirely new estimand or estimating-equation class.

5. Theoretical properties and regularity conditions

Under standard regularity conditions—including strong ignorability, boundedness, Hölder-smoothness of $S_B$4, DNN complexity tuned to approximation versus estimation trade-off, positivity/truncation, and uniform design-consistency of Horvitz–Thompson—the paper states two principal theoretical results (Dai et al., 27 May 2026).

First, if $S_B$5 lies in a composite Hölder class of smoothness $S_B$6 and intrinsic dimension $S_B$7, then with appropriately chosen depth, width, and sparsity, the DNN estimator $S_B$8 satisfies

$S_B$9

The convergence rate is described as near-minimax (Dai et al., 27 May 2026).

Second, under the same conditions, plus positivity/truncation of $i$0, both the DNN-assisted IPW estimator and the deep doubly robust estimator satisfy

$i$1

These results place the DDR framework within the nonparametric-rate literature for deep learning nuisance estimators while preserving the standard consistency claim associated with doubly robust procedures (Dai et al., 27 May 2026).

A broader context appears in related deep doubly robust work for average treatment effects. Rostami et al. define the usual doubly robust ATE estimator

$i$2

and note that first-order bias is $i$3 under general DR theory (Rostami et al., 2021). Although that paper concerns i.i.d. treatment-effect estimation rather than finite-population survey integration, it provides a closely related nuisance-estimation perspective: flexible neural nuisance fits can reduce misspecification bias, but propensity overfitting can jeopardize positivity and inflate variance (Rostami et al., 2021).

6. Empirical evaluation and comparative behavior

The survey-integration study evaluates the proposed estimators in both simulation and real data (Dai et al., 27 May 2026).

In the simulation study, the finite population has $i$4 with four covariates $i$5 and outcome

$i$6

The true selection mechanism $i$7 includes nonlinear terms, including interactions, $i$8, and $i$9. Two misspecification scenarios are considered: “TF,” defined as correct regression plus misspecified linear propensity, and “FF,” defined as both regression and linear-propensity misspecified. The competing estimators are the simple sample mean $x_i\in\mathbb R^r$0, parametric REG, IPW, DR (Chen et al. 2020), DNN-IPW (DIPW), and DNN-DR (DDR). Performance is assessed by relative bias (\%RB) and MSE over 500 replicates (Dai et al., 27 May 2026).

The reported findings are that under nonlinear selection the parametric IPW and DR estimators suffer large bias and MSE when the logistic model is misspecified; DIPW substantially reduces bias; and DDR yields the lowest bias and MSE across all settings, including “FF” (Dai et al., 27 May 2026). The conclusion drawn in the source is that the proposed estimators can improve robustness to parametric propensity-score misspecification, especially when the true selection mechanism is nonlinear (Dai et al., 27 May 2026).

The real-data application uses a nonprobability sample from Pew Research Center 2015 with aggregate $x_i\in\mathbb R^r$1 and a reference sample from BRFSS 2015 with $x_i\in\mathbb R^r$2. The common covariates are age category, gender, race, Hispanic origin, region, marital status, employment, and education. Outcomes include six binary civic-participation or trust variables and one continuous outcome, number of drinking days. The estimated $x_i\in\mathbb R^r$3 values from all methods show that DDR estimates are broadly similar to REG and DR on some outcomes, but differ on others, illustrating sensitivity to the choice of propensity model (Dai et al., 27 May 2026).

A concise summary is given below.

Component	Description	Source
Simulation population	$x_i\in\mathbb R^r$4; four covariates; nonlinear selection	(Dai et al., 27 May 2026)
Misspecification settings	“TF” and “FF”	(Dai et al., 27 May 2026)
Evaluation metrics	relative bias (\%RB) and MSE over 500 replicates	(Dai et al., 27 May 2026)
Real nonprobability sample	Pew Research Center 2015, aggregate $x_i\in\mathbb R^r$5	(Dai et al., 27 May 2026)
Real reference sample	BRFSS 2015, $x_i\in\mathbb R^r$6	(Dai et al., 27 May 2026)

The empirical pattern is consistent with a wider theme in deep doubly robust estimation: neural nuisance models may reduce model misspecification bias, but the quality of the resulting doubly robust estimator still depends on how the nuisance fits affect overlap, weighting stability, and finite-sample variance. Rostami et al. make this point explicitly in ATE estimation, showing that when the treatment model is fit too well by a black-box neural network, $x_i\in\mathbb R^r$7 can approach 0 or 1, violating positivity and elevating variance (Rostami et al., 2021). This suggests that the performance gains of DDR in survey integration are closely tied to flexible but controlled estimation of the sampling score.

7. Relation to adjacent deep doubly robust formulations

The term “deep doubly robust estimator” is used in more than one statistical setting, and the survey-integration formulation should be distinguished from other doubly robust and “doubly doubly robust” constructions.

In the finite-population survey setting, DDR refers to the estimator

$x_i\in\mathbb R^r$8

where the nonprobability-sample sampling score is estimated by a DNN via pseudo-likelihood, and the outcome regression is parametric (Dai et al., 27 May 2026).

In i.i.d. causal inference for the average treatment effect, Rostami et al. study a neural-network implementation of the standard DR estimator in which both the treatment model $x_i\in\mathbb R^r$9 and the outcome regressions $R_i = 1\{i\in S_A\}.$0 are learned by neural networks, with a targeted $R_i = 1\{i\in S_A\}.$1 penalty imposed on the final propensity layer to control variance and preserve overlap (Rostami et al., 2021). Their main emphasis is the bias-variance tradeoff induced by neural propensity overfitting rather than survey integration.

In survival analysis with left-truncated, right-censored data, Pan introduces a “doubly doubly robust” estimator in which one layer of double robustness appears in the survival-loss construction and a second layer appears in the treatment-effect estimator: $R_i = 1\{i\in S_A\}.$2 combined with a standard causal DR estimator for the treatment effect (Pan, 2024). That framework addresses two missing-data problems—counterfactual missingness and missingness due to truncation and censoring—rather than integration of probability and nonprobability surveys (Pan, 2024).

A common misconception is to treat these formulations as interchangeable because they all combine deep learning with doubly robust estimation. The available descriptions indicate otherwise. They share the use of neural nuisance estimation inside doubly robust estimators, but differ in estimands, sampling structures, assumptions, and the precise locus of robustness. The survey-integration DDR is specifically designed for estimating a finite-population mean from a nonprobability sample and a reference probability sample, with the principal deep component located in the sampling-score model (Dai et al., 27 May 2026).