A Note on The Backfitting Estimation of Additive Models

TL;DR

This paper investigates the convergence conditions of the backfitting algorithm for additive models, demonstrating that a weak condition suffices when using Nadaraya-Watson kernel smoothing.

Contribution

It proves that a weak condition guarantees convergence of the backfitting algorithm with Nadaraya-Watson smoothing, improving understanding of its theoretical properties.

Findings

A weak condition ensures convergence with Nadaraya-Watson smoothing.

Convergence proof under less restrictive assumptions.

Clarifies theoretical foundation of backfitting algorithms.

Abstract

The additive model is one of the most popular semiparametric models. The backfitting estimation (Buja, Hastie and Tibshirani, 1989, \textit{Ann. Statist.}) for the model is intuitively easy to understand and theoretically most efficient (Opsomer and Ruppert, 1997, \textit{Ann. Statist.}); its implementation is equivalent to solving simple linear equations. However, convergence of the algorithm is very difficult to investigate and is still unsolved. For bivariate additive models, Opsomer and Ruppert (1997, \textit{Ann. Statist.}) proved the convergence under a very strong condition and conjectured that a much weaker condition is sufficient. In this short note, we show that a weak condition can guarantee the convergence of the backfitting estimation algorithm when the Nadaraya-Watson kernel smoothing is used.