We propose an estimation method of population moments or parameters in “biased sampling data” in which for some units of data, not only the variables of interest but also the covariates have missing observations, and the proportion of “missingness” is unknown. We use auxiliary information such as the distribution of covariates or their moments in random sampling data in order to correct the bias. Moreover, with additional assumptions, we can correct the bias even if we have only the moment information of covariates. The main contribution of this paper is the development of a doubly robust-type estimator for biased sampling data. This method provides a consistent estimator if either the regression function or the assignment mechanism is correctly specified. We prove the consistency and semi-parametric efficiency of the doubly robust estimator. Both the simulation and empirical application results demonstrate that the proposed estimation method is more robust than existing methods.
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