# Doubly Robust Inference when Combining Probability and Non-probability Samples with High-dimensional Data

## TL;DR

This paper develops a method for integrating high-dimensional probability and non-probability samples to make unbiased population inferences, using variable selection and doubly robust estimation techniques.

## Contribution

It introduces a two-step approach combining penalized variable selection with a doubly robust estimator for finite population means in high-dimensional settings.

## Key findings

- The method achieves selection consistency for general probability samples.
- The doubly robust estimator is root-n consistent if either the sampling or outcome model is correct.
- The approach effectively mitigates selection bias in high-dimensional survey data.

## Abstract

Non-probability samples become increasingly popular in survey statistics but may suffer from selection biases that limit the generalizability of results to the target population. We consider integrating a non-probability sample with a probability sample which provides high-dimensional representative covariate information of the target population. We propose a two-step approach for variable selection and finite population inference. In the first step, we use penalized estimating equations with folded-concave penalties to select important variables for the sampling score of selection into the non-probability sample and the outcome model. We show that the penalized estimating equation approach enjoys the selection consistency property for general probability samples. The major technical hurdle is due to the possible dependence of the sample under the finite population framework. To overcome this challenge, we construct martingales which enable us to apply Bernstein concentration inequality for martingales. In the second step, we focus on a doubly robust estimator of the finite population mean and re-estimate the nuisance model parameters by minimizing the asymptotic squared bias of the doubly robust estimator. This estimating strategy mitigates the possible first-step selection error and renders the doubly robust estimator root-n consistent if either the sampling probability or the outcome model is correctly specified.