Spatially adaptive density estimation by localised Haar projections

Florian Gach; Richard Nickl; Vladimir Spokoiny

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 3, page 900-914
  • ISSN: 0246-0203

Abstract

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Given a random sample from some unknown density f 0 : ℝ → [ 0 , ∞ ) we devise Haar wavelet estimators for f 0 with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist.25(1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of f 0 , 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 practise is given.

How to cite

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Gach, Florian, Nickl, Richard, and Spokoiny, Vladimir. "Spatially adaptive density estimation by localised Haar projections." Annales de l'I.H.P. Probabilités et statistiques 49.3 (2013): 900-914. <http://eudml.org/doc/271960>.

@article{Gach2013,
abstract = {Given a random sample from some unknown density $f_\{0\}:\mathbb \{R\}\rightarrow [0,\infty )$ we devise Haar wavelet estimators for $f_\{0\}$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist.25(1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_\{0\}$, 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 practise is given.},
author = {Gach, Florian, Nickl, Richard, Spokoiny, Vladimir},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {spatial adaptation; propagation condition},
language = {eng},
number = {3},
pages = {900-914},
publisher = {Gauthier-Villars},
title = {Spatially adaptive density estimation by localised Haar projections},
url = {http://eudml.org/doc/271960},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Gach, Florian
AU - Nickl, Richard
AU - Spokoiny, Vladimir
TI - Spatially adaptive density estimation by localised Haar projections
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 3
SP - 900
EP - 914
AB - Given a random sample from some unknown density $f_{0}:\mathbb {R}\rightarrow [0,\infty )$ we devise Haar wavelet estimators for $f_{0}$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist.25(1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_{0}$, 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 practise is given.
LA - eng
KW - spatial adaptation; propagation condition
UR - http://eudml.org/doc/271960
ER -

References

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  1. [1] D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika81 (1994) 425–455. Zbl0815.62019MR1311089
  2. [2] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard. Wavelet shrinkage: Asymptopia? J. R. Stat. Soc. Ser. B Stat. Methodol.57 (1995) 301–369. Zbl0827.62035MR1323344
  3. [3] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard. Density estimation by wavelet thresholding. Ann. Statist.24 (1996) 508–539. Zbl0860.62032MR1394974
  4. [4] E. Giné, R. Latala and J. Zinn. Exponential and moment inequalities for U -statistics. In High Dimensional Probability II 13–38. E. Giné, D. Mason and J. A. Wellner (Eds). Birkhäuser Boston, Boston, MA, 2000. Zbl0969.60024MR1857312
  5. [5] E. Giné and R. Nickl. An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation. Probab. Theory Related Fields143 (2009) 569–596. Zbl1160.62032MR2475673
  6. [6] E. Giné and R. Nickl. Uniform limit theorems for wavelet density estimators. Ann. Probab.37 (2009) 1605–1646. Zbl1255.62103MR2546757
  7. [7] E. Giné and R. Nickl. Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli16 (2010) 1137–1163. Zbl1207.62082MR2759172
  8. [8] A. Goldenshluger and O. Lepski. Structural adaptation via 𝕃 p -norm oracle inequalities. Probab. Theory Related Fields143 (2009) 41–71. Zbl1149.62020MR2449122
  9. [9] S. Jaffard. On the Frisch–Parisi conjecture. J. Math. Pures Appl.79 (2000) 525–552. Zbl0963.28009MR1770660
  10. [10] O. V. Lepski, E. Mammen and V. Spokoiny. Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist.25 (1997) 929–947. Zbl0885.62044MR1447734
  11. [11] V. Spokoiny and C. Vial. Parameter tuning in pointwise adaptation using a propagation approach. Ann. Statist.37 (2009) 2783–2807. Zbl1173.62028MR2541447

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