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Variable Kernel Estimates: on the Impossibility of Tuning the Parameters

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High Dimensional Probability II

Part of the book series: Progress in Probability ((PRPR,volume 47))

Abstract

For the standard kernel density estimate, it is known that one can tune the bandwidth such that the expected Li error is within a constant factor of the optimal L1 error (obtained when one is allowed to choose the bandwidth with knowledge of the density). In this paper, we pose the same problem for variable bandwidth kernel estimates where the bandwidths are allowed to depend upon the location. We show in particular that for positive kernels on the real line, for any data-based bandwidth, there exists a density for which the ratio of expected L1 error over optimal L1 error tends to infinity. Thus, the problem of tuning the variable bandwidth in an optimal manner is “too hard”. Moreover, from the class of counterexamples exhibited in the paper, it appears that placing conditions on the densities (monotonicity, convexity, smoothness) does not help.

Research was supported by NSERC Grant A3456 and by FCAR Grant 90-ER-0291.

Research was supported by DIGES Grant PB96-0300.

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Author information

Authors and Affiliations

  1. School of Computer Science, McGill University, Montreal, Canada, H3A 2A7

    Luc Devroye

  2. Department of Economics, Pompeu Fabra University, Ramon Trias Fargas, 25-27, 08005, Barcelona, Spain

    Gárbor Lugosi

Authors
  1. Luc Devroye
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  2. Gárbor Lugosi
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Editor information

Editors and Affiliations

  1. Departments of Mathematics & Statistics, University of Connecticut, Storrs, CT, 06269, USA

    Evarist Giné

  2. Department of Statistics, University of Washington, Seattle, WA, 98195, USA

    Jon A. Wellner

  3. Department of Food and Resource Econ., University of Delaware, Newark, DE, 19717, USA

    David M. Mason

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Devroye, L., Lugosi, G. (2000). Variable Kernel Estimates: on the Impossibility of Tuning the Parameters. In: Giné, E., Mason, D.M., Wellner, J.A. (eds) High Dimensional Probability II. Progress in Probability, vol 47. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1358-1_27

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  • DOI: https://doi.org/10.1007/978-1-4612-1358-1_27

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7111-6

  • Online ISBN: 978-1-4612-1358-1

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