Efficient Estimation of Quadratic Finite Population Functions in the Presence of Auxiliary Information

By viewing quadratic and other second-order finite population functions as totals or means over a derived synthetic finite population, we show that the recently proposed model calibration and pseudoempirical likelihood methods for effective use of auxiliary information from survey data can be readily extended to obtain efficient estimators of quadratic and other second-order finite population functions. In particular, estimation of a finite population variance, covariance, or variance of a linear estimator can be greatly improved when auxiliary information is available. The proposed methods are model assisted in that the resulting estimators are asymptotically design unbiased irrespective of the correctness of a working model but very efficient if the working model is nearly correct. They have a number of attractive features, which include applicability to a general sampling design, incorporation of information on possibly multivariate auxiliary variables, and the ability to entertain linear or nonlinear working models, and they result in nonnegative estimates for certain strictly positive quantities such as variances. Several existing estimators are shown to be special cases of the proposed general methodology under a linear working model.