Density smoothness estimation problem using a wavelet approach
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 130-144.

In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (K. Dziedziul, M. Kucharska, B. Wolnik, Estimation of the smoothness parameter). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes possible for some classes of the density functions.

DOI : 10.1051/ps/2013030
Classification : 62G05, 62G07
Mots-clés : estimation, wavelets, Besov spaces, smoothness parameter 
@article{PS_2014__18__130_0,
     author = {Dziedziul, Karol and \'Cmiel, Bogdan},
     title = {Density smoothness estimation problem using a wavelet approach},
     journal = {ESAIM: Probability and Statistics},
     pages = {130--144},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013030},
     mrnumber = {3143736},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2013030/}
}
TY  - JOUR
AU  - Dziedziul, Karol
AU  - Ćmiel, Bogdan
TI  - Density smoothness estimation problem using a wavelet approach
JO  - ESAIM: Probability and Statistics
PY  - 2014
SP  - 130
EP  - 144
VL  - 18
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2013030/
DO  - 10.1051/ps/2013030
LA  - en
ID  - PS_2014__18__130_0
ER  - 
%0 Journal Article
%A Dziedziul, Karol
%A Ćmiel, Bogdan
%T Density smoothness estimation problem using a wavelet approach
%J ESAIM: Probability and Statistics
%D 2014
%P 130-144
%V 18
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2013030/
%R 10.1051/ps/2013030
%G en
%F PS_2014__18__130_0
Dziedziul, Karol; Ćmiel, Bogdan. Density smoothness estimation problem using a wavelet approach. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 130-144. doi : 10.1051/ps/2013030. http://archive.numdam.org/articles/10.1051/ps/2013030/

[1] E. Belitser and F. Enikeeva, Empirical Bayesian Test of the Smoothness. Math. Methods Stat. 17 (2008) 1-18. | MR | Zbl

[2] A.D. Bull, A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets. Constructive Approximation 37 (2013) 295-309. | MR

[3] A.D. Bull, Honest adaptive confidence bands and self-similar functions. Electron. J. Stat. 6 (2012) 1490-1516. | MR | Zbl

[4] T. Cai, Adaptive Wavelet Estimation: A Block Thresholding and Oracle Inequality Approach. Ann. Stat. 27 (1999) 898-924. | MR | Zbl

[5] T. Cai and M.G. Low, An adaptation theory for nonparametric confidence intervals. Ann. Stat. 32 5 (2004) 1805-1840. | MR | Zbl

[6] T. Cai and M.G. Low, Adaptive confidence balls. Ann. Stat. 34 (2006) 202-228. | MR | Zbl

[7] E. Chicken and T. Cai, Block thresholding for density estimation: local and global adaptivity. J. Multivariate Anal. 95 (2005) 76-106. | MR | Zbl

[8] I. Daubechies, Ten lectures on wavelets. SIAM Philadelphia (1992). | MR | Zbl

[9] D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1996) 879-921. | MR | Zbl

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

[11] K. Dziedziul, M. Kucharska and B. Wolnik, Estimation of the smoothness parameter. J. Nonparametric Stat. 23 (2011) 991-1001. | MR | Zbl

[12] E. Giné and R. Nickl, Confidence bands in density estimation. Ann. Stat. 38 (2010) 1122-1170. | MR | Zbl

[13] A. Gloter and M. Hoffmann, Nonparametric reconstruction of a multifractal function from noisy data. Probab. Theory Relat. Fields 146 (2010) 155187. | MR | Zbl

[14] P. Hall and M.C. Jones, Adaptive M-Estimation in Nonparametric Regression. Ann. Stat. 18 (1990) 1712-1728. | MR | Zbl

[15] W. Härdle, G. Kerkyacharian, D. Picard and A.B. Tsybakov, Wavelets, Approximation and Statistical Applications. Springer-Verlag, New York (1998). | MR | Zbl

[16] M. Hoffmann and R. Nickl, On adaptive inference and confidence bands. Ann. Stat. 39 (2011) 2383-2409. | MR | Zbl

[17] L. Horvath and P. Kokoszka, Change-point detection with non parametric regression. Statistics: A J. Theoret. Appl. Stat. 36 (2002) 9-31. | MR | Zbl

[18] Y. Ingster and N. Stepanova, Estimation and detection of functions from anisotropic Sobolev classes. Electron. J. Stat. 5 (2011) 484-506. | MR | Zbl

[19] S. Jaffard, Conjecture de Frisch et Parisi et généricité des fonctions multifractales. C. R. Acad. Sci. Paris Sér. I Math. 330 4 (2000) 265-270. | MR | Zbl

[20] M.G. Low, On nonparametric confidence intervals. Ann. Stat. 25 (1997) 2547-2554. | MR | Zbl

[21] Y. Meyer, Wavelets and operators. In Cambridge Stud. Advanc. Math. of vol. 37. Translated from the 1990 French original by D.H. Salinger. Cambridge University Press, Cambridge. (1992). | MR | Zbl

[22] S. Ropela, Spline bases in Besov spaces. Bull. Acad. Pol. Sci. Serie Math. astr. Phys. 24 (1976) 319-325. | MR | Zbl

[23] S.J. Sheather and M.C. Jones, A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation. J. Royal Stat. Soc. Ser. B. 53 (1991) 683-690. | MR | Zbl

Cité par Sources :