Variable bandwidth and one-step local M-estimator

Jianqing Fan, Jiancheng Jiang

Research output: Contribution to journalArticlepeer-review

44 Scopus citations


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 least-squares 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 one-step 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.

Original languageEnglish (US)
Pages (from-to)65-81
Number of pages17
JournalScience in China, Series A: Mathematics, Physics, Astronomy
Issue number1
StatePublished - Jan 2000
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Local regression
  • M-estimator
  • Nonparametric estimation
  • One-step
  • Robustness
  • Variable bandwidth


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