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Spatially Adaptive Density Estimation by Localised Haar Projections

Author

Listed:
  • Florian Gach
  • Richard Nickl
  • Vladimir Spokoiny

Abstract

Given a random sample from some unknown density f0 : R → [0;∞) we devise Haar wavelet estimators for f0 with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen, and Spokoiny (1997, Ann. Statist.)). We show that these estimators adapt to spatially heterogeneous smoothness of f0, simultaneously for every point x in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under the idealised assumption that the true density is locally constant in a neighborhood of the point x of estimation, and an information theoretic justi cation of this practice is given.

Suggested Citation

  • Florian Gach & Richard Nickl & Vladimir Spokoiny, 2011. "Spatially Adaptive Density Estimation by Localised Haar Projections," SFB 649 Discussion Papers SFB649DP2011-078, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    • Handle: RePEc:hum:wpaper:sfb649dp2011-078
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2011-078.pdf
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    Citations

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    Cited by:

    1. Raffaele Fiocco, 2012. "Competition and regulation with product differentiation," Journal of Regulatory Economics, Springer, vol. 42(3), pages 287-307, December.
    2. Patrick Cheridito & Ulrich Horst & Michael Kupper & Traian A. Pirvu, 2011. "Equilibrium Pricing in Incomplete Markets under Translation Invariant Preferences," SFB 649 Discussion Papers SFB649DP2011-083, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Alena MyÅ¡iÄ ková & Song Song & Piotr Majer & Peter N.C. Mohr & Hauke R. Heekeren & Wolfgang K. Härdle, 2011. "Risk Patterns and Correlated Brain Activities. Multidimensional statistical analysis of fMRI data with application to risk patterns," SFB 649 Discussion Papers SFB649DP2011-085, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Ulrich Horst & Michael Kupper & Andrea Macrina & Christoph Mainberger, 2011. "Continuous Equilibrium under Base Preferences and Attainable Initial Endowments," SFB 649 Discussion Papers SFB649DP2011-082, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

    More about this item

    Keywords

    spatially inhomogeneous smoothness; bandwidth choice; propagation approach;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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