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Annals of the Institute of Statistical Mathematics

, Volume 61, Issue 3, pp 663–690 | Cite as

Efficient and fast spline-backfitted kernel smoothing of additive models

  • Jing WangEmail author
  • Lijian Yang
Article

Abstract

A great deal of effort has been devoted to the inference of additive model in the last decade. Among existing procedures, the kernel type are too costly to implement for high dimensions or large sample sizes, while the spline type provide no asymptotic distribution or uniform convergence. We propose a one step backfitting estimator of the component function in an additive regression model, using spline estimators in the first stage followed by kernel/local linear estimators. Under weak conditions, the proposed estimator’s pointwise distribution is asymptotically equivalent to an univariate kernel/local linear estimator, hence the dimension is effectively reduced to one at any point. This dimension reduction holds uniformly over an interval under assumptions of normal errors. Monte Carlo evidence supports the asymptotic results for dimensions ranging from low to very high, and sample sizes ranging from moderate to large. The proposed confidence band is applied to the Boston housing data for linearity diagnosis.

Keywords

Bandwidths B spline Knots Local linear estimator Nadaraya-Watson estimator Nonparametric regression 

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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2007

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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