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Probability Theory and Related Fields

, Volume 135, Issue 3, pp 335–362 | Cite as

Propagation-Separation Approach for Local Likelihood Estimation

  • Jörg Polzehl
  • Vladimir Spokoiny
Article

Abstract

The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, density, Poisson and binary response model. The method extends the adaptive weights smoothing (AWS) procedure introduced in Polzehl and Spokoiny (2000) in context of image denoising. The main idea of the method is to describe a greatest possible local neighborhood of every design point X i in which the local parametric assumption is justified by the data. The method is especially powerful for model functions having large homogeneous regions and sharp discontinuities. The performance of the proposed procedure is illustrated by numerical examples for density estimation and classification. We also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm. This includes the ``propagation'' property which particularly yields the root-n consistency of the resulting estimate in the homogeneous case. We also state an ``oracle'' result which implies rate optimality of the estimate under usual smoothness conditions and a ``separation'' result which explains the sensitivity of the method to structural changes.

Mathematics Subject Classification (2000)

62G05 Secondary: 62G07 62G08 62G32 62H30 

Key words or phrases

Adaptive weights Local likelihood Exponential family Propagation Separation Density estimation Classification 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Weierstrass-InstituteBerlinGermany

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