Estimation of the finite population distribution function using a global penalized calibration method

Abstract

Auxiliary information $${\varvec{x}}$$ is commonly used in survey sampling at the estimation stage. We propose an estimator of the finite population distribution function $$F_{y}(t)$$ when $${\varvec{x}}$$ is available for all units in the population and related to the study variable y by a superpopulation model. The new estimator integrates ideas from model calibration and penalized calibration. Calibration estimates of $$F_{y}(t)$$ with the weights satisfying benchmark constraints on the fitted values distribution function $$\hat{F}_{\hat{y}}=F_{\hat{y}}$$ on a set of fixed values of t can be found in the literature. Alternatively, our proposal $$\hat{F}_{y\omega }$$ seeks an estimator taking into account a global distance $$D(\hat{F}_{\hat{y}\omega },F_{\hat{y}})$$ between $$\hat{F}_{\hat{y}\omega }$$ and $${F}_{\hat{y}},$$ and a penalty parameter $$\alpha $$ that assesses the importance of this term in the objective function. The weights are explicitly obtained for the $$L^2$$ distance and conditions are given so that $$\hat{F}_{y\omega }$$ to be a distribution function. In this case $$\hat{F}_{y\omega }$$ can also be used to estimate the population quantiles. Moreover, results on the asymptotic unbiasedness and the asymptotic variance of $$\hat{F}_{y\omega }$$ , for a fixed $$\alpha $$ , are obtained. The results of a simulation study, designed to compare the proposed estimator to other existing ones, reveal that its performance is quite competitive.