Variance estimation and asymptotic confidence bands for the mean estimator of sampled functional data with high entropy unequal probability sampling designs

TL;DR

This paper extends variance estimation techniques for the mean of sampled functional data under high entropy unequal probability sampling, enabling the construction of asymptotic confidence bands with proven convergence properties.

Contribution

It generalizes the H"ajek variance formula to functional data and provides a uniformly convergent estimator with convergence rates, facilitating confidence band construction.

Findings

The variance estimator converges uniformly to the true variance function.

Confidence bands achieve asymptotic correct coverage via Gaussian process simulation.

Application to electricity consumption data demonstrates practical effectiveness.

Abstract

For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson estimator can be approximated by the H\'ajek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz-Thompson estimator of the mean function. Rates of convergence to the true variance function are given for the rejective sampling. We deduce, under conditions on the entropy of the sampling design, that it is possible to build confidence bands whose coverage is asymptotically the desired one via simulation of Gaussian processes with variance function given by the H\'ajek formula. Finally, the accuracy of the proposed variance estimator is evaluated on samples of electricity consumption data measured every half an hour over a period of one week.