AStA Advances in Statistical Analysis

, Volume 103, Issue 1, pp 1–35 | Cite as

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

  • J. A. Mayor-Gallego
  • J. L. Moreno-RebolloEmail author
  • M. D. Jiménez-Gamero
Original Paper


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.


Auxiliary information Model-assisted approach Sample survey Penalized calibration estimator 



The authors thank the anonymous reviewers for constructive comments. M.D. Jiménez-Gamero acknowledges financial support from grant MTM2014-55966-P of the Spanish Ministry of Economy and Competitiveness, and grant MTM2017-89422-P of the Spanish Ministry of Economy, Industry and Competitiveness, ERDF support included.

Supplementary material


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of SevilleSevilleSpain

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