Uniform convergence and asymptotic confidence bands for model-assisted estimators of the mean of sampled functional data

Hervé Cardot (1) , Camelia Goga (1) , Pauline Lardin (1, 2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 EDF R&D - EDF R&D

When 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 is 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.

Mots clés: calibration; covariance function; functional linear model; GREG; Hájek estimator; Horvitz-Thompson estimator; linear interpolation.; linear interpolation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Hervé Cardot, Camelia Goga, Pauline Lardin. Uniform convergence and asymptotic confidence bands for model-assisted estimators of the mean of sampled functional data. 2012. ⟨hal-00691943v1⟩