On Robust Recursive Nonparametric Curve Estimation

  • Eduard Belitser
  • Sara van de Geer
Conference paper
Part of the Progress in Probability book series (PRPR, volume 47)

Abstract

Suppose we observe X k = θ(x k ) + ξ k . The function θ: [0,1] → ℝ, is assumed to belong a priori to a given nonparametric smoothness class, the ξ k ’s are independent identically distributed random variables with zero medians. The only prior information about the distribution of the noise is that it belongs to a rather wide class. The assumptions describing this class include cases in which no moments of the noises exist, so that linear estimation methods (for example, kernel methods) can not be applied. We propose a robust estimator based on a stochastic approximation procedure and derive its rate of convergence, as the frequency of observations n tends to infinity, in almost sure as well as in mean square sense, uniformly over the smoothness class Finally, we discuss a multivariate formulation of the problem, a robust nonparametric M-estimator (the least deviations estimator), the so called penalized estimator, and the case when the noises are not necessarily identically distributed.

Keywords

Entropy Filtration Dian Terion 

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References

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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Eduard Belitser
    • 1
  • Sara van de Geer
    • 1
  1. 1.Mathematical InstituteLeiden UniversityLeidenThe Netherlands

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