Nonparametrics for Complex Parameters Estimation in Finite Population. Application to the Estimation of the Gini Index with Survey Data
Camelia Goga, Anne Ruiz-Gazen
IMB, Universite de Bourgogne, Dijon, France; TSE-IMT, Universite Toulouse 1 Capitole, Toulouse, France

The problem of estimating with good precision complex parameters such as quantiles, Gini indices or other measures of inequalities is particularly crucial nowadays. Estimating totals or means by taking into account auxiliary information has been extensively studied in the finite population context. In the model-assisted approach, both linear and non-parametric models have been considered and non-parametric models have shown their superiority in terms of precision if the linear model of the survey on the auxiliary variable is misspecified. Concerning complex parameters, the literature is much less abundant and is mostly dedicated to quantiles. In the present paper, we propose a general class of estimators for complex parameters that take into account a complete auxiliary information through a non-parametric model. We derive their asymptotic variance and propose some variance estimators in a very general framework by using the influence function and the linearized variable concepts. The interesting point is that the gain in efficiency when taking into account the auxiliary information for a complex parameter depends on the prediction quality of the model for the linearized variable of the survey variable on the auxiliary variable. Because linearized variables may be quite complex, linear models are likely not to perform well and are outperformed by non-parametrics even if the survey variable is linearly related with the auxiliary one. The theory is developed in a general non-parametrics framework but many details are given for the B-splines estimators including practical implementation and guidelines for choosing the smoothing parameters. Several calibration properties of the B-splines estimators are exhibited such as a model-calibration property. Furthermore, the good asymptotic and finite-sample performances of the proposed estimators are illustrated on two real data sets. In particular, point and confidence intervals estimation of the Gini index are considered for television audience measurements where the gain of using B-splines estimators is clearly demonstrated.

Keywords: B-splines regression; calibration estimators; linearization; influence function

Biography: Anne Ruiz-Gazen is Profesor in Statistics at the University Toulouse 1 Capitole from Toulouse, France. Her research themes concern multivariate statistics, robust statistics and survey sampling theory among others.