Nonparametric local polynomial regression for functional covariates

TL;DR

This work advances nonparametric regression with functional covariates by constructing a locally polynomial estimator that operates in a finite-dimensional projection of the functional space. By leveraging Fréchet differentiability and supersmoothness, it derives a general asymptotic upper bound on the pointwise error and proves that polynomial convergence rates are attainable under suitable basis choice and tuning parameters $J$ and $K$. The results rely on careful control of small-ball probabilities and spectral properties of the projected design, yielding rates $|\hat{g}(x)-g(x)|^2=O_P(n^{-\kappa})$ for some $\kappa>0$ under favorable conditions, including exponential eigenvalue decay and strong smoothness. This has practical impact for functional data analysis where traditional nonparametric rates are typically subpolynomial, demonstrating that polynomial rates can be achieved without overly restrictive linearity assumptions. The methodology and proofs provide a rigorous framework for extending local polynomial regression to infinite-dimensional covariates.

Abstract

We consider nonparametric regression with functional covariates, that is, they are elements of an infinite-dimensional Hilbert space. A locally polynomial estimator is constructed, where an orthonormal basis and various tuning parameters remain to be selected. We provide a general asymptotic upper bound on the estimation error and show that this procedure achieves polynomial convergence rates under appropriate tuning and supersmoothness of the regression function. Such polynomial convergence rates have usually been considered to be non-attainable in nonparametric functional regression without any additional strong structural constraints such as linearity of the regression function.

Theorems & Definitions (9)

• Lemma 2.1
• Lemma 3.1
• Example 1
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Proposition 3.2
• Theorem 3.1
• proof : Proof of Lemma \ref{['L:bounddim']}