Penalized spline estimation of principal components for sparse functional data: rates of convergence

TL;DR

This paper analyzes the convergence rates of penalized spline estimators for multiple principal components in sparse functional data, revealing how various factors influence their asymptotic behavior and optimality.

Contribution

It provides a comprehensive theoretical framework for the convergence rates of penalized spline estimators in sparse functional principal component analysis, including classification into seven scenarios.

Findings

Identifies seven distinct asymptotic scenarios for estimator behavior.

Characterizes conditions for achieving minimax optimal convergence rates.

Highlights the impact of smoothness, spline degree, and penalty parameters on estimator performance.

Abstract

This paper gives a comprehensive treatment of the convergence rates of penalized spline estimators for simultaneously estimating several leading principal component functions, when the functional data is sparsely observed. The penalized spline estimators are defined as the solution of a penalized empirical risk minimization problem, where the loss function belongs to a general class of loss functions motivated by the matrix Bregman divergence, and the penalty term is the integrated squared derivative. The theory reveals that the asymptotic behavior of penalized spline estimators depends on the interesting interplay between several factors, i.e., the smoothness of the unknown functions, the spline degree, the spline knot number, the penalty order, and the penalty parameter. The theory also classifies the asymptotic behavior into seven scenarios and characterizes whether and how the minimax optimal rates of convergence are achievable in each scenario.