This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA' estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach.

Revised version for the Electronic Journal of StatisticsInternational audienceWhen the study variable is functional and storage capacities are limited or transmission costs are high, selecting with survey sampling techniques a small fraction of the observations is an interesting alternative to signal compression techniques, particularly when the goal is the estimation of simple quantities such as means or totals. We extend, in this functional framework, model-assisted estimators with linear regression models that can take account of auxiliary variables whose totals over the population are known. We first show, under weak hypotheses on the sampling design and the regularity of the trajectories, that the estimator of the mean function as well as its variance estimator are uniformly consistent. Then, under additional assumptions, we prove a functional central limit theorem and we assess rigorously a fast technique based on simulations of Gaussian processes which is employed to build asymptotic confidence bands. The accuracy of the variance function estimator is evaluated on a real dataset of sampled electricity consumption curves measured every half an hour over a period of one week

&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