Nonparametric Depth and Quantile Regression for Functional Data

TL;DR

This paper develops nonparametric regression methods using spatial depth and quantiles for functional data, enabling analysis of how covariates influence the entire distribution of functional responses, including tails, with applications to economic and health datasets.

Contribution

It introduces novel nonparametric depth and quantile regression techniques for functional data, extending classical concepts to infinite-dimensional settings with asymptotic analysis.

Findings

Effective detection of heteroscedasticity in functional regression

Asymptotic properties depend on small ball probabilities

Applied to economic and health datasets with meaningful insights

Abstract

We investigate nonparametric regression methods based on spatial depth and quantiles when the response and the covariate are both functions. As in classical quantile regression for finite dimensional data, regression techniques developed here provide insight into the influence of the functional covariate on different parts, like the center as well as the tails, of the conditional distribution of the functional response. Depth and quantile based nonparametric regressions are useful to detect heteroscedasticity in functional regression. We derive the asymptotic behaviour of nonparametric depth and quantile regression estimates, which depend on the small ball probabilities in the covariate space. Our nonparametric regression procedures are used to analyse a dataset about the influence of per capita GDP on saving rates for 125 countries, and another dataset on the effects of per capita net disposable income on the sale of cigarettes in some states in the US.