While the additive model is a popular nonparametric regression method, many of its theoretical properties are not well understood, especially when the backfitting algorithm is used for computation of the the estimators. This article explores those properties when the additive model is fitted by local polynomial regression. Sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given. Asymptotic approximations to the bias and the variance of a homoskedastic bivariate additive model with local polynomial terms are computed. This model is shown to have the same rate of convergence as that of univariate local polynomial regression. We also investigate the estimation of derivatives of the additive component functions.

When additive models with more than two covariates are fitted with the backfitting algorithm proposed by Buja et al. [2], the lack of explicit expressions for the estimators makes study of their theoretical properties cumbersome. Recursion provides a convenient way to extend existing theoretical results for bivariate additive models to models of arbitrary dimension. In the case of local polynomial regression smoothers, recursive asymptotic bias and variance expressions for the backfitting estimators are derived. The estimators are shown to achieve the same rate of convergence as those of univariate local polynomial regression. In the case of independence between the covariates, non-recursive bias and variance expressions, as well as the asymptotically optimal values for the bandwidth parameters, are provided. 1 Introduction The additive model, originally suggested by Friedman and Stuetzle [4], assumes that the conditional expectation function of the dependent variable can be written a...

Nonparametric regression techniques are often sensitive to the presence of correlation in the errors. The practical consequences of this sensitivity are explained, including the breakdown of several popular data-driven smoothing parameter selection methods. We review the existing literature in kernel regression, smoothing splines and wavelet regression under correlation, both for short-range and long-range dependence. Extensions to random design, higher dimensional models and adaptive estimation are discussed.

Generalized additive models are a popular class of multivariate nonparametric regression models, due in large part to the ease of use of the local scoring estimation algorithm. However, the theoretical properties of the local scoring estimator are poorly understood. In this article, we propose a local likelihood estimator for generalized additive models that is closely related to the local scoring estimator tted by local polynomial regression. We derive the statistical properties of the estimator and show that it achieves the same asymptotic convergence rate as a one-dimensional local polynomial regression estimator. We also propose a wild bootstrap estimator for calculating pointwise condence intervals for the additive component functions. The practical behavior of the proposed estimator is illustrated through simulation experiments and an example. Keywords: backtting, bootstrapping, generalized additive models, local likelihood, local polynomial regression, local scoring, wild bo...

The smooth backfitting introduced by Mammen, Linton and Nielsen [Ann. Statist. 27 (1999) 1443–1490] is a promising technique to fit additive regression models and is known to achieve the oracle efficiency bound. In this paper, we propose and discuss three fully automated bandwidth selection methods for smooth backfitting in additive models. The first one is a penalized least squares approach which is based on higher-order stochastic expansions for the residual sums of squares of the smooth backfitting estimates. The other two are plug-in bandwidth selectors which rely on approximations of the average squared errors and whose utility is restricted to local linear fitting. The large sample properties of these bandwidth selection methods are given. Their finite sample properties are also compared through simulation experiments. 1. Introduction. Nonparametric

This paper tackles the problem of model complexity in the context of additive models. Several methods have been proposed to estimate smoothing parameters, as well as to perform variable selection. Nevertheless,

In this paper a new smooth backfitting estimate is proposed for additive regression models. The estimate has the simple structure of Nadaraya–Watson smooth backfitting but at the same time achieves the oracle property of local linear smooth backfitting. Each component is estimated with the same asymptotic accuracy as if the other components were known. 1. Introduction. In

Automated bandwidth selection methods for nonparametric regression break down in the presence of correlated errors. While this problem has been previously studied in the context of kernel regression, the results to date have only been applicable to univariate observations following an equidistant design. This article addresses the problem for local linear regression and considers both univariate and bivariate observations following either a fixed or random design. In the bivariate case, we analyze both the general bivariate model and the additive model. In this more general setting, we show that when the errors are correlated, the asymptotically optimal bandwidth depends on the integrated covariance function. Nonparametric plug-in bandwidth estimators which take this effect into account are proposed, and estimators of regression functionals, variance and integrated covariance function are developed. In particular, an estimate of the integrated covariance function is constructed by esti...

We explore additive models that combine both parametric and nonparametric terms and propose a p n-consistent backfitting estimator for the parametric component of the model. The theoretical properties of the estimator are developed for the case with a single nonparametric term and extended to an arbitrary number of nonparametric additive terms. An estimator for the optimal bandwidth making minimal use of asymptotic expressions for bias and variance is proposed, and a fast implementation algorithm for model fitting and bandwidth selection is developed. The practical behavior of the estimator and bandwidth selection is illustrated by simulation experiments. Key Words: local polynomial regression, bandwidth selection, EBBS, partially linear model. 1 Introduction Additive models are a popular and flexible class of nonparametric regression methods (Hastie and Tibshirani (1990)), which assume that the conditional mean function can be represented as E(Y jZ 1 ; : : : ; ZD ) = m 1 (Z 1 ) + ...

Abstract not found