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Suppose that the nite population consists of N identi able units. Associated with the ith unit are the study variable, y i , and a vector of auxiliary variables, x i . The values x 1 1 x 2 1 : : : 1 x N are known for the entire population (i.e., complete) but y i is known only if the ith unit is selected in the sample. One of the fundamental questions is how to effectively use the complete auxiliary information at the estimation stage. In this article, a uni ed model-assisted framework has been attempted using a proposed model-calibration technique. The proposed model-calibration estimators can handle any linear or nonlinear working models and reduce to the conventional calibration estimators of Deville and Särndal and/or the generalized regression estimators in the linear model case. The pseudoempirical maximum likelihood estimator of Chen and Sitter, when used in this setting, gives an estimator that is asymptotically equivalent to the model-calibration estimator but with positive weights. Some existing estimators using auxiliary information are reexamined under this framework. The estimation of the nite population distribution function, using complete auxiliary information, is also considered, and estimators based on a general model are presented. Results of a limited simulation study on the performance of the proposed estimators are reported.

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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.KEY WORDS Design-based inference, inclusion probability, missing at random, probability sampling, propensity score, regression modeling, variance estimation.

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