Local linear smoothing in additive models as data projection

TL;DR

This paper presents a new interpretation of local linear smooth backfitting in additive models as a data projection, simplifying the understanding and analysis of its properties.

Contribution

It introduces a projection-based interpretation of local linear smooth backfitting, making the complex asymptotic analysis more accessible and intuitive.

Findings

Simplifies the mathematical analysis of smooth backfitting

Provides a clearer understanding of estimator properties

Achieves optimal convergence rates under smoothness conditions

Abstract

We discuss local linear smooth backfitting for additive non-parametric models. This procedure is well known for achieving optimal convergence rates under appropriate smoothness conditions. In particular, it allows for the estimation of each component of an additive model with the same asymptotic accuracy as if the other components were known. The asymptotic discussion of local linear smooth backfitting is rather complex because typically an overwhelming notation is required for a detailed discussion. In this paper we interpret the local linear smooth backfitting estimator as a projection of the data onto a linear space with a suitably chosen semi-norm. This approach simplifies both the mathematical discussion as well as the intuitive understanding of properties of this version of smooth backfitting.