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Spatially adaptive density estimation by localised Haar projections

Author

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  • Gach, Florian
  • Nickl, Richard
  • Spokoiny, Vladimir

Abstract

Given a random sample from some unknown density f0 : R → [0;∞) we devise Haar wavelet estimators for fo 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 justification of this practice is given.

Suggested Citation

  • Gach, Florian & Nickl, Richard & Spokoiny, Vladimir, 2011. "Spatially adaptive density estimation by localised Haar projections," SFB 649 Discussion Papers 2011-078, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    • Handle: RePEc:zbw:sfb649:sfb649dp2011-078
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    File URL: https://www.econstor.eu/bitstream/10419/56625/1/675475082.pdf
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    Cited by:

    1. repec:hum:wpaper:sfb649dp2011-083 is not listed on IDEAS
    2. Horst, Ulrich & Kupper, Michael & Macrina, Andrea & Mainberger, Christoph, 2011. "Continuous equilibrium under base preferences and attainable initial endowments," SFB 649 Discussion Papers 2011-082, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    3. repec:hum:wpaper:sfb649dp2011-082 is not listed on IDEAS
    4. repec:hum:wpaper:sfb649dp2011-084 is not listed on IDEAS
    5. Cheridito, Patrick & Horst, Ulrich & Kupper, Michael & Pirvu, Traian A., 2011. "Equilibrium pricing in incomplete markets under translation invariant preferences," SFB 649 Discussion Papers 2011-083, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    6. Myšičková, Alena & Song, Song & Majer, Piotr & Mohr, Peter N. C. & Heekeren, Hauke R. & Härdle, Wolfgang Karl, 2011. "Risk patterns and correlated brain activities: Multidimensional statistical analysis of fMRI data with application to risk patterns," SFB 649 Discussion Papers 2011-085, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. repec:hum:wpaper:sfb649dp2011-085 is not listed on IDEAS
    8. Fiocco, Raffaele, 2011. "Competition and regulation in a differentiated good market," SFB 649 Discussion Papers 2011-084, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.

    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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