Second-order asymptotic theory for calibration estimators in sampling and missing-data problems

Consider three different but related problems with auxiliary information: infinite population sampling or Monte Carlo with control variates, missing response with explanatory variables, and Poisson and rejective sampling with auxiliary variables. We demonstrate unified regression and likelihood estimators and study their second-order properties. The likelihood estimators are second-order unbiased but the regression estimators are not. For the missing-data problem and survey sampling, no estimator studied always has the smallest second-order variance even after bias correction. However, the calibrated likelihood estimator and bias-corrected, calibrated regression estimator are second-order more efficient than other bias-corrected estimators if a linear model holds for the conditional expectation of the response or study variable given explanatory or auxiliary variables.