Penalized Spline M-Estimators for Discretely Sampled Functional Data: Existence and Asymptotics

TL;DR

This paper develops a broad penalized spline M-estimation framework for discretely sampled functional data, establishing existence, asymptotic properties, and optimal convergence rates, with practical robustness and computational advantages.

Contribution

It extends existing non-robust estimators to a general penalized M-estimation framework with new theoretical tools, including non-asymptotic existence and localization lemmas.

Findings

Optimal convergence rates are established under mild assumptions.

Parametric rates are achievable with discretely sampled data.

Numerical experiments show the proposed estimators are robust and computationally efficient.

Abstract

Location estimation is a central problem in functional data analysis. In this paper, we investigate penalized spline estimators of location for discretely sampled functional data under a broad class of convex loss functions. Our framework generalizes and extends previously derived results for non-robust estimators to a broad penalized M-estimation framework. The analysis is built on two general-purpose non-asymptotic theoretical tools: (i) a non-asymptotic existence result for penalized spline estimators under minimal design conditions, and (ii) a localization lemma that captures both stochastic variability and approximation error. Under mild assumptions, we establish optimal convergence rates and identify the small- and large-knot regimes, along with the critical breakpoint known from penalized spline theory in nonparametric regression. Our results imply that parametric rates are attainable even with discretely sampled data and numerical experiments demonstrate that our estimators match or exceed the robustness of competing estimators while being considerably more computationally attractive.