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Minimizing L1 distance in nonparametric density estimation

Author

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  • Hall, Peter
  • Wand, Matthew P.

Abstract

We construct a simple algorithm, based on Newton's method, which permits asymptotic minimization of L1 distance for nonparametric density estimators. The technique is applicable to multivariate kernel estimators, multivariate histogram estimators, and smoothed histogram estimators such as frequency polygons. It has an "adaptive" or "data-driven" version. We show theoretically that both theoretical and adaptive forms of the algorithm do indeed minimize asymptotic L1 distance. Then we apply the algorithm to derive concise formulae for asymptotically optimal smoothing parameters. We also give numerical examples of applications of the adaptive algorithm.

Suggested Citation

  • Hall, Peter & Wand, Matthew P., 1988. "Minimizing L1 distance in nonparametric density estimation," Journal of Multivariate Analysis, Elsevier, vol. 26(1), pages 59-88, July.
    • Handle: RePEc:eee:jmvana:v:26:y:1988:i:1:p:59-88
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    Citations

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    Cited by:

    1. Jizba, Petr & Korbel, Jan, 2014. "Multifractal diffusion entropy analysis: Optimal bin width of probability histograms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 438-458.
    2. Wang, Xiao-Feng & Ye, Deping, 2015. "Conditional density estimation in measurement error problems," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 38-50.
    3. Davies, P. Laurie & Gather, Ursula & Nordman, Daniel & Weinert, Henrike, 2007. "Constructing a regular histogram : a comparison of methods," Technical Reports 2007,14, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    4. Klemelä, Jussi, 2000. "Estimation of Densities and Derivatives of Densities with Directional Data," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 18-40, April.
    5. Petr Jizba & Jan Korbel, 2014. "Multifractal Diffusion Entropy Analysis: Optimal Bin Width of Probability Histograms," Papers 1401.3316, arXiv.org, revised Mar 2014.
    6. Ouimet, Frédéric & Tolosana-Delgado, Raimon, 2022. "Asymptotic properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 187(C).

    More about this item

    Keywords

    asymptotic optimality histogram estimator kernel estimator L1 distance nonparametric density estimator;

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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