Optimal inference for the mean of random functions
Omar Kassi, Valentin Patilea
TL;DR
This work addresses inference for the mean function $\mu$ of random functions on $\mathcal{T}=[0,1]^D$ under random design and heteroscedastic noise by introducing a Fourier-series–based estimator built from de La Vallée Poussin projections. A fast control-variate linear-integration scheme is used to estimate Fourier coefficients, achieving minimax rates over Hölder classes and providing non-asymptotic $L^2$ risk bounds with explicit constants, plus non-asymptotic Gaussian approximations for coefficient vectors to enable pointwise and uniform confidence sets. The approach is made adaptive by plug-in estimators for the Hölder regularity $\alpha_\mu$, with concentration bounds ensuring reliable selection of the smoothing level and Hölder constant. The results cover both dense and sparse sampling regimes through a phase-transition in rates, and yield practical inference tools (confidence bands and regions) that respect the randomness of the design and heteroscedastic noise. Overall, the paper advances adaptive, nonparametric mean-function inference for random functional data in high dimensions using a Fourier-analytic framework with non-asymptotic guarantees.
Abstract
We study estimation and inference for the mean of real-valued random functions defined on a hypercube. The independent random functions are observed on a discrete, random subset of design points, possibly with heteroscedastic noise. We propose a novel optimal-rate estimator based on Fourier series expansions and establish a sharp non-asymptotic error bound in $L^2-$norm. Additionally, we derive a non-asymptotic Gaussian approximation bound for our estimated Fourier coefficients. Pointwise and uniform confidence sets are constructed. Our approach is made adaptive by a plug-in estimator for the Hölder regularity of the mean function, for which we derive non-asymptotic concentration bounds.