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Science in China Series A: Mathematics

, Volume 43, Issue 1, pp 65–81 | Cite as

Variable bandwidth and one-step local M-estimator

  • Jianqing Fan
  • Jiancheng Jiang
Article

Abstract

A robust version of local linear regression smoothers augmented with variable bandwidth is studied. The proposed method inherits the advantages of local polynomial regression and overcomes the shortcoming of lack of robustness of leastsquares techniques. The use of variable bandwidth enhances the flexibility of the resulting local M-estimators and makes them possible to cope well with spatially inhomogeneous curves, heteroscedastic errors and nonuniform design densities. Under appropriate regularity conditions, it is shown that the proposed estimators exist and are asymptotically normal. Based on the robust estimation equation, one-step local M-estimators are introduced to reduce computational burden. It is demonstrated that the one-step local M-estimators share the same asymptotic distributions as the fully iterative M-estimators, as long as the initial estimators are good enough. In other words, the onestep local M-estimators reduce significantly the computation cost of the fully iterative M-estimators without deteriorating their performance. This fact is also illustrated via simulations.

Keywords

local regression M-estimator nonparametric estimation one-step robustness variable bandwidth 

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Copyright information

© Science in China Press 2000

Authors and Affiliations

  • Jianqing Fan
    • 1
  • Jiancheng Jiang
    • 2
  1. 1.University of North Carolina at Chapel Hill and Chinese University of Hong KongHong KongChina
  2. 2.Department of Probability & StatisticsPeking UniversityBeijingChina

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