Additive regression with general imperfect variables

TL;DR

This paper develops an additive regression model for Hilbert-space-valued responses and imperfect predictors, providing theoretical properties and demonstrating its versatility through simulations and real data applications.

Contribution

It introduces a novel additive regression framework accommodating imperfectly observed Hilbert-space-valued responses and predictors, with comprehensive theoretical analysis.

Findings

The estimator has desirable non-asymptotic properties.

Simulation studies confirm the estimator's effectiveness.

Real data applications demonstrate practical utility.

Abstract

In this paper, we study an additive model where the response variable is Hilbert-space-valued and predictors are multivariate Euclidean, and both are possibly imperfectly observed. Considering Hilbert-space-valued responses allows to cover Euclidean, compositional, functional and density-valued variables. By treating imperfect responses, we can cover functional variables taking values in a Riemannian manifold and the case where only a random sample from a density-valued response is available. This treatment can also be applied in semiparametric regression. Dealing with imperfect predictors allows us to cover various principal component and singular component scores obtained from Hilbert-space-valued variables. For the estimation of the additive model having such variables, we use the smooth backfitting method. We provide full non-asymptotic and asymptotic properties of our regression estimator and present its wide applications via several simulation studies and real data applications.