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March, 1988 Robust Nonparametric Regression with Simultaneous Scale Curve Estimation
W. Hardle, A. B. Tsybakov
Ann. Statist. 16(1): 120-135 (March, 1988). DOI: 10.1214/aos/1176350694


Let $\{X_i, Y_i\}^n_{i=1} \subset \mathbb{R}^d \times \mathbb{R}$ be independent identically distributed random variables. If the conditional distribution $F(y \mid x)$ can be parametrized by $F(y \mid x) = F_0((y - m(x))/\sigma(x))$ with a fixed and known distribution $F_0$, the regression curve $m(x)$ and scale curve $\sigma(x)$ could be estimated by some parametric method. More generally, we assume that $F$ is unknown and consider nonparametric simultaneous $M$-type estimates of the unknown functions $m(x)$ and $\sigma(x)$, using kernel estimators for the conditional distribution function $F(y \mid x)$. We show pointwise consistency and asymptotic normality of these estimates. The rate of convergence is optimal in the sense of Stone (1980). The asymptotic bias term of this robust estimate turns out to be the same as for the linear Nadaraya-Watson kernel estimate.


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W. Hardle. A. B. Tsybakov. "Robust Nonparametric Regression with Simultaneous Scale Curve Estimation." Ann. Statist. 16 (1) 120 - 135, March, 1988.


Published: March, 1988
First available in Project Euclid: 12 April 2007

zbMATH: 0668.62025
MathSciNet: MR924860
Digital Object Identifier: 10.1214/aos/1176350694

Primary: 62G05

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 1 • March, 1988
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