Asymptotic Properties of Backfitting Estimators

Abstract

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. Academic Press

Additive and Generalized Additive Models

“…regression splines) and smoothing splines but not for general linear smoothers. Opsomer and Ruppert (1997) and Opsomer (1996) derived similar results for local linear regression, a nonprojection smoother, under rather strong conditions on the smoothing matrices. Here we relate back tting to standard iterative procedures, i.e.…”

“…He derived recursive expressions for the back tting estimator. Also for d > 2 the existence and the uniqueness of this estimator depend on the characteristics of the pairwise products of the smoother matrices: A d-variate additive model with smoother matrices S 1 : : : S d will converge to an unique solution, if for some matrix norm where denotes the s block consisting of a n n identity matrix and zero elements otherwise in a matrix E , so thatĝ = E A ;1 By (provided the inverse of A for equation system (1.5) exists for more details see Opsomer, 1996). Thus the index corresponds to the 's block of the solution vectorĝ.…”

Efficient and fast spline-backfitted kernel smoothing of additive models

“…Hastie and Tibshirani (1990) proposed backfitting estimators for functions {m α (x α )} d α=1 without theoretical justifications, while Opsomer and Ruppert (1997) offered partial asymptotic results for the case of d = 2 under some strong assumptions. Opsomer (2000) extended the theoretical results to a general case with more than two covariates. Mammen et al (1999) proposed a modified backfitting algorithm with nice theoretical properties, which was implemented in Nielsen and Sperlich (2005) and called smooth backfitting estimator.…”

Feasible estimation in generalized structured models

“…We consider our generalized varying coefficient model to be more appropriate in two ways: it has a clear interpretable and intuitive structure, and it has a lower dimensionality. Apart from that, the statistical behavior of their estimator is unknown but supposed to be less efficient, compare Mammen and Nielsen (2003) and Opsomer (2000).…”

“…Hastie and Tibshirani (1990) introduced estimation procedures based on the backfitting algorithm for additive models, a particular member of our model class. Later, Opsomer and Ruppert (1997) and Opsomer (2000) developed asymptotic theory for this backfitting estimator. Several alternatives have been proposed to estimate (generalized) additive models, marginal impacts and interaction terms, see Binder and Tutz (2008) for various popular spline based methods, and Sperlich et al (2002) for marginal integration.…”