## Electronic Journal of Statistics

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

#### Abstract

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 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.

#### Article information

Source
Electron. J. Statist. Volume 7 (2013), 562-596.

Dates
First available in Project Euclid: 14 March 2013

http://projecteuclid.org/euclid.ejs/1363268498

Digital Object Identifier
doi:10.1214/13-EJS779

Mathematical Reviews number (MathSciNet)
MR3035266

Zentralblatt MATH identifier
1336.62043

Subjects
Primary: 62L20: Stochastic approximation
Secondary: 60F05: Central limit and other weak theorems

#### Citation

Cardot, Hervé; Goga, Camelia; Lardin, Pauline. Uniform convergence and asymptotic confidence bands for model-assisted estimators of the mean of sampled functional data. Electron. J. Statist. 7 (2013), 562--596. doi:10.1214/13-EJS779. http://projecteuclid.org/euclid.ejs/1363268498.

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