Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs For partial linear model Y = X τ β 0 + g 0(T) + ε with unknown β 0 ∈¸ R d and an unknown smooth function g 0, this paper considers the Huber-Dutter estimators of β 0, scale σ for the errors and the function g 0 respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β 0 and σ are asymptotically normal with convergence rate n -1/2 and the B-spline Huber-Dutter estimator of g 0 achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β 0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 2 (2005) 257–268 Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs 1 2 3 Xing-wei Tong ,Heng-jian Cui ,Hui Zhao 1,2 Department of Mathematics, Beijing Normal University, Beijing 100875, China (E-mail: hjcui@bnu.edu.cn) Department of Statistics, Central China Normal University, Wuhan 430079, China τ d Abstract For partial linear model Y = X β + g (T)+ ε with unknown β ∈ R and an unknown smooth 0 0 0 function g , this paper considers the Huber-Dutter estimators of β ,scale σ for the errors and the function 0 0 g respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown −1/2 that the Huber-Dutter estimators of β and σ are asymptotically normal with convergence rate n and the B-spline Huber-Dutter estimator of g achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study. Keywords Huber-dutter estimator, partial linear model, m-estimator, optimal convergence rate, B-spline function 2000 MR Subject Classification 62G20, 62G35 1 Introduction Consider the partial linear model: y = x β + g (t )+ ε,i =1,··· ,n, (1.1) i 0 0 i i where y is real-valued, x ∈ R , t ∈ [0, 1], β is a d-vector of unknown parameters, g is an i i i 0 0 unknown smooth function, and ε ,i =1,··· ,n are i.i.d. random errors. Since it contains a nonparametric component, this model is more flexible than the usual standard linear models [4] and has been used by Engle et al. in studyng the relation between weather and electricity sales. In the last two decades, a number of robust...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0234-y
Publisher site
See Article on Publisher Site

Abstract

For partial linear model Y = X τ β 0 + g 0(T) + ε with unknown β 0 ∈¸ R d and an unknown smooth function g 0, this paper considers the Huber-Dutter estimators of β 0, scale σ for the errors and the function g 0 respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β 0 and σ are asymptotically normal with convergence rate n -1/2 and the B-spline Huber-Dutter estimator of g 0 achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β 0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

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