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

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.