The Annals of Statistics

Smooth backfitting for errors-in-variables additive models

Kyunghee Han and Byeong U. Park

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In this work, we develop a new smooth backfitting method and theory for estimating additive nonparametric regression models when the covariates are contaminated by measurement errors. For this, we devise a new kernel function that suitably deconvolutes the bias due to measurement errors as well as renders a projection interpretation to the resulting estimator in the space of additive functions. The deconvolution property and the projection interpretation are essential for a successful solution of the problem. We prove that the method based on the new kernel weighting scheme achieves the optimal rate of convergence in one-dimensional deconvolution problems when the smoothness of measurement error distribution is less than a threshold value. We find that the speed of convergence is slower than the univariate rate when the smoothness of measurement error distribution is above the threshold, but it is still much faster than the optimal rate in multivariate deconvolution problems. The theory requires a deliberate analysis of the nonnegligible effects of measurement errors being propagated to other additive components through backfitting operation. We present the finite sample performance of the deconvolution smooth backfitting estimators that confirms our theoretical findings.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 2216-2250.

Received: November 2016
Revised: July 2017
First available in Project Euclid: 17 August 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Nonparametric additive regression smooth backfitting errors-in-variables models deconvolution kernel smoothing


Han, Kyunghee; Park, Byeong U. Smooth backfitting for errors-in-variables additive models. Ann. Statist. 46 (2018), no. 5, 2216--2250. doi:10.1214/17-AOS1617.

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Supplemental materials

  • Supplement to “Smooth backfitting for errors-in-variables additive models”. The supplement contains proofs of Lemmas 5.1, 5.2 and 5.3. It also gives proofs of (3.1), (5.22) and (5.24).