Réduction de la variance dans les sondages en présence d'information auxiliarie: Une approache non paramétrique par splines de régression

Abstract

&I. 33, No. 2,2005, Pages 163-180 La revue canadiennede sratisrique 1 63 Reduction de la variance dans les sondages en presence d'information auxiliaire : une approche non parametrique par splines de regression Camelia GOGA MSC 2000: Primary 62D05; secondary 62608.Rkssumt : L'auteure traite de la prise en compte par un modtle non param6trique de I'infonnation auxiliaire dans le but d'am6liorer I'estimateur d'un total. Elle introduit un nouveau type d'estimateur assist6 par un modkle fond6 sur les splines de r6gression. L'estimateur obtenu est une somme lin6aire pond6r6e des valeurs de la variable d'intt&t avec des poids calks sur les totaux des fonctions B-splines. L'auteure montre que I'estimateur est asymptotiquement sans biais et convergent sous des conditions indtpendantes de la validit6 du modtle. Elle donne une approximation pour la variance sous le plan de sondage et dtmontre que la variance anticip6e est asymptotiquement Bquivalente h la borne infkrieure de Godambe-Joshi. Elle v6rifie en outre le bon cornportement de cet estimateur par voie de simulation.Variance reduction in surveys with auxiliary information: a nonparametric approach involving regression splines Abstract: The author considers the use of auxiliary information available at population level to improve the estimation of finite population totals. She introduces a new type of model-assisted estimator based on nonpararnetric regression splines. The estimator is a weighted linear combination of the study variable with weights calibrated to the B-splines known population totals. The author shows that the estimator is asymptotically design-unbiased and consistent under conditions which do not require the superpopulation model to be correct. She proposes a design-based variance approximation and shows that the anticipated variance is asymptotically equivalent to the Godambe-Joshi lower bound. She also shows through simulations that the estimator has good propenies. 1 64 GOGA Vol. 33, No. 2

Efficient Estimation of Non-Linear Finite Population Parameters by Using Non-Parametrics

“…One way of guarding against model failure is to use non-parametric regression which does not require a predefined parametric mathematical expression for f . Breidt and Opsomer (2000) proposed local linear estimators and Breidt et al (2005) and Goga (2005) used non-parametric spline regression. The unknown f function is approximated by the projection of the population vector y U = .y k / k∈U onto different basis functions, such as the basis of truncated qth-degree polynomials in Breidt et al (2005) and the B-spline basis in Goga (2005).…”

“…Breidt and Opsomer (2000) proposed local linear estimators and Breidt et al (2005) and Goga (2005) used non-parametric spline regression. The unknown f function is approximated by the projection of the population vector y U = .y k / k∈U onto different basis functions, such as the basis of truncated qth-degree polynomials in Breidt et al (2005) and the B-spline basis in Goga (2005). In what follows, we briefly recall the definition and the main asymptotic properties of non-parametric model-assisted estimators for finite population totals (see also Breidt and Opsomer (2009)).…”

“…This pseudoestimator is still design unbiased but it is model biased because non-parametric estimators f y,k are biased for f.z k / (Sarda and Vieu, 2000). Nevertheless, under supplementary assumptions (Breidt and Opsomer, 2000;Goga, 2005), the bias under the model vanishes asymptotically to 0 when the population and the sample sizes go to ∞. The unknown quantitiesf y,k are usually obtained by least squares methods (ordinary, weighted or penalized) and we may writẽ…”

“…Therefore, non-parametric models should be preferred to estimate non-linear parameters Φ. Recent work already employs non-parametric models to estimate totals (Breidt and Opsomer, 2000;Breidt et al, 2005;Goga, 2005). The use of non-parametrics prevents model failure; however, the improvement over parametric estimation for totals and means may not be sufficiently significant to justify the supplemental difficulties of implementing non-parametric methodology.…”

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

“…5-7); generalised regression estimators under any parametric linear model (Cassel, Särndal, and Wretman 1976;Särndal, Swensson, and Wretman 1992), calibration estimators (Deville and Särndal 1992;Wu and Sitter 2001), and empirical likelihood estimators (Chen and Sitter 1999;Chen, Sitter, and Wu 2002). The desire for flexible specification of f (·) has led to the consideration of local polynomial regression (Breidt and Opsomer 2000), penalised splines (Breidt et al 2005), neural networks (Montanari and Ranalli 2005b), regression splines (Goga 2005), additive and generalised additive models (Opsomer, Breidt, Moisen, and Kauermann 2007; Wang and Wang 2011), among many others. See Särndal (2007) for some general review of model-assisted estimation and Montanari and Ranalli (2005a) and Breidt and Opsomer (2009) for nonparametric and semiparametric model-assisted methods.…”