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Kernel density estimators: convergence in distribution for weighted sup-norms
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  • Published: 03 March 2004

Kernel density estimators: convergence in distribution for weighted sup-norms

  • Evarist Giné1,
  • Vladimir Koltchinskii2 &
  • Lyudmila Sakhanenko3 

Probability Theory and Related Fields volume 130, pages 167–198 (2004)Cite this article

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Abstract.

Let f n denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) be a positive continuous function such that Under natural smoothness conditions, necessary and sufficient conditions for the sequence \scriptstyle{t\inR/}\big|\Psi(t)(fn(t)-Efn(t))\big| (properly centered and normalized) to converge in distribution to the double exponential law are obtained. The proof is based on Gaussian approximation and a (new) limit theorem for weighted sup-norms of a stationary Gaussian process. This extends well known results of Bickel and Rosenblatt to the case of weighted sup-norms, with the sup taken over the whole line. In addition, all other possible limit distributions of the above sequence are identified (subject to some regularity assumptions).

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Author information

Authors and Affiliations

  1. Departments of Mathematics and Statistics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Evarist Giné

  2. Department of Mathematics and Statistics, University of New Mexico Albuquerque, NM, 87131-1141, USA

    Vladimir Koltchinskii

  3. Department of Statistics and Probability, Michigan State University East Lansing, MI, 48824-1027, USA

    Lyudmila Sakhanenko

Authors
  1. Evarist Giné
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  2. Vladimir Koltchinskii
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  3. Lyudmila Sakhanenko
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Corresponding author

Correspondence to Vladimir Koltchinskii.

Additional information

1. Research partially supported by NSF Grant No. DMS-0070382.

2. Research partially supported by NSA Grant No. MDA904-02-1-0075 and NSF Grant No. DMS-0304861.

Mathematics Subject Classification (2000):Primary 62G07; secondary 62G20, 60F15.

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Cite this article

Giné, E., Koltchinskii, V. & Sakhanenko, L. Kernel density estimators: convergence in distribution for weighted sup-norms. Probab. Theory Relat. Fields 130, 167–198 (2004). https://doi.org/10.1007/s00440-004-0339-x

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  • Received: 02 July 2002

  • Revised: 24 December 2003

  • Published: 03 March 2004

  • Issue Date: October 2004

  • DOI: https://doi.org/10.1007/s00440-004-0339-x

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Key words and phrases:

  • Kernel density estimator
  • Convergence in distribution
  • Weighted sup-norm
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