  We establish a general framework for statistical inferences with
non-probability survey samples when relevant auxiliary information is available
from a probability survey sample. We develop a rigorous procedure for
estimating the propensity scores for units in the non-probability sample, and
construct doubly robust estimators for the finite population mean. Variance
estimation is discussed under the proposed framework. Results from simulation
studies show the robustness and the efficiency of our proposed estimators as
compared to existing methods. The proposed method is used to analyze a
non-probability survey sample collected by the Pew Research Center with
auxiliary information from the Behavioral Risk Factor Surveillance System and
the Current Population Survey. Our results illustrate a general approach to
inference with non-probability samples and highlight the importance and
usefulness of auxiliary information from probability survey samples.
