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

TL;DR

This paper develops uniform convergence results and constructs asymptotic confidence bands for model-assisted estimators of the mean function in sampled functional data, addressing challenges in signal compression and transmission costs.

Contribution

It extends model-assisted estimation techniques to functional data, providing uniform consistency, a functional CLT, and practical confidence bands under survey sampling.

Findings

Estimator and variance estimator are uniformly consistent.

A functional CLT is established under additional assumptions.

Confidence bands are effectively constructed using Gaussian process simulations.

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.