Multivariate kernel regression in vector and product metric spaces

TL;DR

This paper establishes limit properties and asymptotic normality of multivariate kernel regression estimators in diverse metric spaces, including cases with complex distributions like fractals and factor structures, with practical applications demonstrated.

Contribution

It extends kernel regression theory to multivariate and functional data in general metric spaces, allowing for complex distributions and demonstrating improved convergence rates.

Findings

Faster convergence with singular distributions like fractals.

Asymptotic normality holds under broad conditions.

Empirical results confirm theoretical rate improvements.

Abstract

This paper derives limit properties of nonparametric kernel regression estimators without requiring existence of density for regressors in $\mathbb{R}^{q}.$ In functional regression limit properties are established for multivariate functional regression. The rate and asymptotic normality for the Nadaraya-Watson (NW) estimator is established for distributions of regressors in $\mathbb{R}^{q}$ that allow for mass points, factor structure, multicollinearity and nonlinear dependence, as well as fractal distribution; when bounded density exists we provide statistical guarantees for the standard rate and the asymptotic normality without requiring smoothness. We demonstrate faster convergence associated with dimension reducing types of singularity, such as a fractal distribution or a factor structure in the regressors. The paper extends asymptotic normality of kernel functional regression to multivariate regression over a product of any number of metric spaces. Finite sample evidence confirms rate improvement due to singularity in regression over $\mathbb{R}^{q}.$ For functional regression the simulations underline the importance of accounting for multiple functional regressors. We demonstrate the applicability and advantages of the NW estimator in our empirical study, which reexamines the job training program evaluation based on the LaLonde data.