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 Fox News Voter Analysis (FNVA), conducted in partnership with the Associated Press, provides a comprehensive look at voting behavior, opinions and preferences as America votes. It is based on surveys conducted in all 50 states by NORC at the University of Chicago, as well as actual voting results by county collected by The AP. The FNVA survey encompasses interviews with an estimated 140,000 registered voters and is conducted Oct. 26 to Nov. 3, and continues through the end of voting on Election Day. Both voters and nonvoters are interviewed to provide a full picture of the election, including why some Americans voted while others stayed at home. FNVA combines respondent interviews from three data sources: (1) a random sample of registered voters drawn from state voter files; (2) a sample of self-identified registered voters conducted using NORC's probability-based panel, which is designed to be representative of the U.S. population; and (3) a sample of self-identified registered voters selected from nonprobability online panels.