Adaptive inference for the mean of a stochastic process in functional data

TL;DR

This paper introduces a fully data-driven, adaptive method for estimating the mean function of a stochastic process from noisy, discretely observed data, providing confidence bands that adapt to unknown regularity.

Contribution

It develops novel thresholded least squares estimators with data-driven thresholds and basis selection, achieving adaptive estimation and confidence bands without covariance estimation.

Findings

Method performs well in simulations, showing robustness to noise.

Estimates adapt to unknown sparsity and regularity of the mean function.

Constructs uniform confidence bands that are easy to compute.

Abstract

This paper proposes and analyzes fully data driven methods for inference about the mean function of a stochastic process from a sample of independent trajectories of the process, observed at discrete time points and corrupted by additive random error. The proposed method uses thresholded least squares estimators relative to an approximating function basis. The variable threshold levels are estimated from the data and the basis is chosen via cross-validation from a library of bases. The resulting estimates adapt to the unknown sparsity of the mean function relative to the selected approximating basis, both in terms of the mean squared error and supremum norm. These results are based on novel oracle inequalities. In addition, uniform confidence bands for the mean function of the process are constructed. The bands also adapt to the unknown regularity of the mean function, are easy to compute, and do not require explicit estimation of the covariance operator of the process. The simulation study that complements the theoretical results shows that the new method performs very well in practice, and is robust against large variations introduced by the random error terms.