Spatially adaptive density estimation by localised Haar projections
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 900-914.

A partir d’un échantillon d’une loi de densité f 0 :[0,), nous construisons des estimateurs par ondelettes de Haar de f 0 , dont les niveaux de résolution varient et sont construits à partir de tests localisés (comme dans l’article Lepski (Ann. Statist. 25 (1997) 927-947)). Nous montrons que ces estimateurs satisfont une inégalité oracle adaptive par rapport à la régularité potentiellement hétérogène de f 0 , simultanément pour tout point x dans un intervalle donné, en norme infinie. Les constantes de seuillage utilisées dans les procédures de test peuvent être choisies en pratique en supposant de manière idéalisée que la vraie densité est localement constante dans un voisinage du point x considéré, pratique que nous justifions par un argument de théorie de l’information.

Given a random sample from some unknown density f 0 :[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.

DOI : 10.1214/12-AIHP485
Classification : 62G05
Mots-clés : spatial adaptation, propagation condition
@article{AIHPB_2013__49_3_900_0,
     author = {Gach, Florian and Nickl, Richard and Spokoiny, Vladimir},
     title = {Spatially adaptive density estimation by localised {Haar} projections},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {900--914},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {3},
     year = {2013},
     doi = {10.1214/12-AIHP485},
     mrnumber = {3112439},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/12-AIHP485/}
}
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
SP  - 900
EP  - 914
VL  - 49
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/12-AIHP485/
DO  - 10.1214/12-AIHP485
LA  - en
ID  - AIHPB_2013__49_3_900_0
ER  - 
%0 Journal Article
%A Gach, Florian
%A Nickl, Richard
%A Spokoiny, Vladimir
%T Spatially adaptive density estimation by localised Haar projections
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2013
%P 900-914
%V 49
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/12-AIHP485/
%R 10.1214/12-AIHP485
%G en
%F AIHPB_2013__49_3_900_0
Gach, Florian; Nickl, Richard; Spokoiny, Vladimir. Spatially adaptive density estimation by localised Haar projections. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 900-914. doi : 10.1214/12-AIHP485. http://archive.numdam.org/articles/10.1214/12-AIHP485/

[1] D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 (1994) 425-455. | MR | Zbl

[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. | MR | Zbl

[3] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard. Density estimation by wavelet thresholding. Ann. Statist. 24 (1996) 508-539. | MR | Zbl

[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. | MR | Zbl

[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 Fields 143 (2009) 569-596. | MR | Zbl

[6] E. Giné and R. Nickl. Uniform limit theorems for wavelet density estimators. Ann. Probab. 37 (2009) 1605-1646. | MR | Zbl

[7] E. Giné and R. Nickl. Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli 16 (2010) 1137-1163. | MR | Zbl

[8] A. Goldenshluger and O. Lepski. Structural adaptation via 𝕃 p -norm oracle inequalities. Probab. Theory Related Fields 143 (2009) 41-71. | MR | Zbl

[9] S. Jaffard. On the Frisch-Parisi conjecture. J. Math. Pures Appl. 79 (2000) 525-552. | MR | Zbl

[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. | MR | Zbl

[11] V. Spokoiny and C. Vial. Parameter tuning in pointwise adaptation using a propagation approach. Ann. Statist. 37 (2009) 2783-2807. | MR | Zbl

Cité par Sources :