We show that the model-calibration estimator for the finite population mean, which was proposed by Wu & Sitter (2001) through an intuitive argument, is optimal among a class of calibration estimators. We also present optimal calibration estimators for the finite population distribution function, the population variance, the variance of a linear estimator and other quadratic finite population functions under a unified framework. The proposed calibration estimators are optimal under the true model but remain design consistent even if the working model is misspecified. A limited simulation study shows that the improvement of these optimal estimators over the conventional ones can be substantial. The question of when and how auxiliary information can be used for both the estimation of the population mean using a generalised regression estimator and the estimation of its variance through calibration is addressed clearly under the proposed general methodology. Some fundamental issues in using auxiliary information from survey data are also addressed in the context of optimal estimation. Copyright Biometrika Trust 2003, Oxford University Press.
