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While probability sampling has been considered the gold standard of survey methods, nonprobability sampling is increasingly popular due to its convenience and low cost. However, nonprobability samples can lead to biased estimates due to the unknown nature of the underlying selection mechanism. In this article, we propose parametric and semiparametric approaches to integrate probability and nonprobability samples using common ancillary variables observed in both samples. In the parametric approach, the joint distribution of ancillary variables is assumed to follow the latent Gaussian copula model, which is flexible to accommodate both categorical and continuous variables. In contrast, the semiparametric approach requires no assumptions about the distribution of ancillary variables. In addition, logistic regression is used to model the mechanism by which population units enter the nonprobability sample. The unknown parameters in the copula model are estimated through the pseudo maximum likelihood approach. The logistic regression model is estimated by maximizing the sample likelihood constructed from the nonprobability sample. The proposed method is evaluated in the context of estimating the population mean. Our simulation results show that the proposed method is able to correct the selection bias in the nonprobability sample by consistently estimating the underlying inclusion mechanism. By incorporating additional information in the nonprobability sample, the combined method can estimate the population mean more efficiently than using the probability sample alone. A real-data application is provided to illustrate the practical use of the proposed method.

 

Nonparametric model-assisted estimators have been proposed to improve estimates of finite population parameters. Flexible nonparametric models provide more reliable estimators when a parametric model is misspecified. In this article, we propose an information criterion to select appropriate auxiliary variables to use in an additive model-assisted method. We approximate the additive nonparametric components using polynomial splines and extend the Bayesian Information Criterion (BIC) for finite populations. By removing irrelevant auxiliary variables, our method reduces model complexity and decreases estimator variance. We establish that the proposed BIC is asymptotically consistent in selecting the important explanatory variables when the true model is additive without interactions, a result supported by our numerical study. Our proposed method is easier to implement and better justified theoretically than the existing method proposed in the literature.