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Optimal global rates of convergence ofM-estimates for multivariate nonparametric regression

Optimal global rates of convergence ofM-estimates for multivariate nonparametric regression Consider the nonparametric regression modelY=go(T)+u, whereY is real-valued,u is a random error,T is a randomd-vector of explanatory variables ranging over a nondegenerated-dimensional compact setC, andgo(·) is the unknown smooth regression function, which ism (≥0) times continuously differentiable and itsmth partial derivatives $$\frac{{\partial ^m g0(t)}}{{\partial ^i 1 \cdot \cdot \cdot \partial ^i d}}$$ satisfy the Hölder condition with exponentγε(0,1], wherei 1, ...,i d are nonnegative integers satisfying Σ k =1/d i k =m. The piecewise polynomial estimator ofgo based onM-estimates is considered. It is proved that the rate of convergence of the underlying estimator is $$O_p \left( {n^{ - \tfrac{{m + \gamma }}{{2(m + \gamma ) + d}}} } \right)$$ under certain regular conditions, which is the optimal global rate of convergence of least square estimates for nonparametric regression studied in [10–11]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Optimal global rates of convergence ofM-estimates for multivariate nonparametric regression

Shi, Peide; Wang, Xueren
Acta Mathematicae Applicatae Sinica , Volume 11 (2) – Jul 14, 2005
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References (15)

Publisher
Springer Journals
Copyright
Copyright © 1995 by Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02013149
Publisher site
See Article on Publisher Site

Abstract

Consider the nonparametric regression modelY=go(T)+u, whereY is real-valued,u is a random error,T is a randomd-vector of explanatory variables ranging over a nondegenerated-dimensional compact setC, andgo(·) is the unknown smooth regression function, which ism (≥0) times continuously differentiable and itsmth partial derivatives $$\frac{{\partial ^m g0(t)}}{{\partial ^i 1 \cdot \cdot \cdot \partial ^i d}}$$ satisfy the Hölder condition with exponentγε(0,1], wherei 1, ...,i d are nonnegative integers satisfying Σ k =1/d i k =m. The piecewise polynomial estimator ofgo based onM-estimates is considered. It is proved that the rate of convergence of the underlying estimator is $$O_p \left( {n^{ - \tfrac{{m + \gamma }}{{2(m + \gamma ) + d}}} } \right)$$ under certain regular conditions, which is the optimal global rate of convergence of least square estimates for nonparametric regression studied in [10–11].

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Jul 14, 2005

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