Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling

TL;DR

This paper develops Horvitz-Thompson estimators for functional data in large datasets, providing asymptotic confidence bands and optimal stratified sampling strategies to improve estimation accuracy and efficiency.

Contribution

It introduces a new Horvitz-Thompson estimator for mean trajectories, proves its consistency and asymptotic normality, and derives optimal stratified sampling rules for functional data.

Findings

Stratified sampling improves estimator accuracy.

Confidence bands are narrower with stratification.

Method applied to electricity consumption data.

Abstract

When dealing with very large datasets of functional data, survey sampling approaches are useful in order to obtain estimators of simple functional quantities, without being obliged to store all the data. We propose here a Horvitz--Thompson estimator of the mean trajectory. In the context of a superpopulation framework, we prove under mild regularity conditions that we obtain uniformly consistent estimators of the mean function and of its variance function. With additional assumptions on the sampling design we state a functional Central Limit Theorem and deduce asymptotic confidence bands. Stratified sampling is studied in detail, and we also obtain a functional version of the usual optimal allocation rule considering a mean variance criterion. These techniques are illustrated by means of a test population of N=18902 electricity meters for which we have individual electricity consumption measures every 30 minutes over one week. We show that stratification can substantially improve both the accuracy of the estimators and reduce the width of the global confidence bands compared to simple random sampling without replacement.