Survey design asymptotics for the model-assisted penalised spline regression estimator

Abstract

The total of a study variable in a finite population may be estimated using data from a complex survey via Horvitz–Thompson estimation. If additional auxiliary information is available, then efficiency is often improved via model-assisted survey regression estimation. Semiparametric models based on penalised spline regression are particularly attractive in this context, as they lead to natural extensions of classical survey regression estimators. Existing theory for the model-assisted penalised spline regression estimator does not account for the setting in which the number of knots is large relative to sample size. This gap is addressed by considering survey design asymptotics for the model-assisted penalised spline survey regression estimator, as the finite population size, sample size, and number of knots all increase to infinity. Conditions on the sequence of designs are developed under which the estimator is consistent for the finite population total and its variance is consistently estimated.