Model-assisted estimation from randomized response data with high-dimensional auxiliary information

Model-assisted estimation is widely used in survey sampling to produce efficient estimations for finite population parameters such as the population total, mean, etc. The idea is to leverage available auxiliary data through a working model that connects the auxiliary variables with the variable of interest. The regression estimator is perhaps the most popular example of model-assisted estimators where the working model employed is a linear regression model. However, the regression estimator may not perform very well when the available auxiliary data is noisy, i.e. has many noise variables which is typical when working with high-dimensional auxiliary data. Recently, model-assisted estimation has been expanded to include other statistical learning techniques that allow for variable selection. In the context of direct response surveys, the survey-weighted LASSO was developed to enable model-assisted estimation with high-dimensional auxiliary data where only some auxiliary variables may be significant to the study variable. In this paper, we consider estimating the finite population mean from a randomized response survey where only randomized responses are available for the study variable due to its confidential/sensitive nature along with complete non-randomized high-dimensional auxiliary information. We extend the survey-weighted LASSO technique to perform model-assisted estimation from a randomized response survey with high-dimensional auxiliary data. We provide asymptotics that bridges the model-assisted survey-weighted LASSO estimator from direct response to the randomized response context. Using Monte Carlo simulations and a real data example, we compare the performance of the randomized response model-assisted survey-weighted LASSO estimator to other estimators, including the randomized response Horvitz-Thompson and generalized regression estimators under different working model structures and sampling designs. We also introduce and study a bootstrap variance estimator for estimating the sampling variance of the proposed mean estimator.