Optimal inference for the mean of random functions

2025/04/15 by Omar Kassi, Kassi, Omar, Valentin Patilea +1 · 1 citation
Mathematics · #Statistical Methods and Inference #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2504.11025

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 L²-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.

Citations

• Structural adaptation via directional regularity: rate accelerated estimation in multivariate functional data
• Functional Data Analysis: An Introduction and Recent Developments
• Functional Data Analysis: An Introduction and Recent Developments
• Learning the regularity of multivariate functional data
• Adaptive functional principal components analysis
• Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence
• Learning the smoothness of noisy curves with application to online curve estimation
• A functional-data approach to the Argo data
• Simultaneous Confidence Corridors for Mean Functions in Functional Data Analysis of Imaging Data
• Fast and Fair Simultaneous Confidence Bands for Functional Parameters
• Functional data analysis in the Banach space of continuous functions
• A Geometric Approach to Confidence Regions and Bands for Functional Parameters
• On the measure of Voronoi cells
• Control functionals for Monte Carlo integration
• Some Results on the Complexity of Numerical Integration
• On adaptive inference and confidence bands
• Optimal estimation of the mean function based on discretely sampled functional data: Phase transition
• Simultaneous confidence bands for nonparametric regression with functional data
• Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition
• Multivariate normal approximation using exchangeable pairs

Cited by

• Nonparametric Estimation from Correlated Copies of a Drifted 2nd-Order Process

Related

• Worst-case recovery guarantees for least squares approximation using random samples
• Nearly Optimal Robust Mean Estimation via Empirical Characteristic Function
• Dimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector
• The optimal sub-Gaussian normalisation for randomised monotone functions
• Near-optimal mean estimators with respect to general norms