Smooth backfitting in generalized additive models

TL;DR

This paper introduces a new likelihood-based method for fitting generalized additive models using smooth backfitting, providing theoretical properties and numerical comparisons with existing estimators.

Contribution

It proposes a novel likelihood approach with an iterative smooth backfitting algorithm for generalized additive models, achieving oracle-like bias and variance.

Findings

The method achieves the same bias and variance as the oracle estimator.

The iterative algorithm converges reliably under specified conditions.

Numerical comparisons show competitive performance with existing estimators.

Abstract

Generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non-Gaussian responses including binary and count data. In this paper, a new likelihood approach for fitting generalized additive models is proposed. It aims to maximize a smoothed likelihood. The additive functions are estimated by solving a system of nonlinear integral equations. An iterative algorithm based on smooth backfitting is developed from the Newton--Kantorovich theorem. Asymptotic properties of the estimator and convergence of the algorithm are discussed. It is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components. Numerical comparison with the recently proposed two-stage estimator [Ann. Statist. 32 (2004) 2412--2443] is also made.