Smooth backfitting for errors-in-variables additive models

Abstract

We study nonparametric additive regression models when noisy covariates are observed within measurement errors. Based on deconvolution techniques, we construct an iterative algorithm for smooth backfitting of additive function in the presence of errors-in-variables. We show that the smooth backfitting achieves univariate accuracy of the standard deconvolution for estimating each component function under certain conditions. Deconvolving noise on backfitting is confined into negligible magnitude that rate of convergence of the proposed estimator is accelerated when the smoothness of measurement errors falls into a certain range. We also present finite sample performance of the deconvolution smooth backfitting in comparison with a naive application of the standard smooth backfitting ignoring measurement errors. Monte Carlo simulation is demonstrated that our method gives smaller mean integrated squared errors than the naive one in average.

Estimation of Errors-in-Variables Partially Linear Additive Models

“…This showsD Now, we come to the termD 14 . From Theorems 2 and 3 in Han and Park (2017), we get that, for 1 ≤ ≤ d,…”

“…Unfortunately, it turns out that this does not work since the resulting normalized deconvolution kernel does not have the unbiased scoring property, so that it fails to deconvolute the effects of measurement errors. Recently, Han and Park (2017) introduced a special kernel scheme that has both the properties of normalization and unbiased scoring, which we adopt here. Let φ f for a function f denote the Fourier transform of f and φ V for a random variable V the characteristic function of V .…”

“…For z ∈ [2h, 1 − 2h] one can show that φ K (·; z) = φ K , the Fourier transform of the baseline kernel K. The special kernel function of Han and Park (2017) is given by…”

“…The conditions ( They enable us to obtain an inequality enveloping K h that we use to get bounds for terms involving K h , see Lemma 5.1 in Han and Park (2017).…”

“…To get an empirical version of (1.3) we estimate η and ξ using a kernel smoothing technique. In particular, we use the smooth backfitting technique of Mammen et al (1999) and the smoothed normalized deconvolution kernel of Han and Park (2017). To the best of our knowledge, this work is the first that studies kernel estimation of the model (1.1) based on the observation of the contaminated predictors X * and Z * .…”