Optimal inference for the mean of random functions

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 H\"older regularity of the mean function, for which we derive non-asymptotic concentration bounds.