  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.
