Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T02:04:20.522Z Has data issue: false hasContentIssue false
  • English
  • Français

Density smoothness estimation problem usinga wavelet approach

Published online by Cambridge University Press:  28 November 2013

Karol Dziedziul  and
Bogdan Ćmiel
Karol Dziedziul
Affiliation:
Faculty of Applied Mathematics, Gdańsk University of Technology, ul. G. Narutowicza 11/12, 80-233 Gdańsk, Poland. [email protected]
Bogdan Ćmiel
Affiliation:
Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Cracov, Poland; [email protected]
Get access

Abstract

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.

Keywords

EstimationwaveletsBesov spacessmoothness parameter
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 130 - 144
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belitser, E. and Enikeeva, F., Empirical Bayesian Test of the Smoothness. Math. Methods Stat. 17 (2008) 1–18. Google Scholar
Bull, A.D., A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets. Constructive Approximation 37 (2013) 295–309. Google Scholar
Bull, A.D., Honest adaptive confidence bands and self-similar functions. Electron. J. Stat. 6 (2012) 1490–1516. Google Scholar
Cai, T., Adaptive Wavelet Estimation: A Block Thresholding and Oracle Inequality Approach. Ann. Stat. 27 (1999) 898–924. Google Scholar
Cai, T. and Low, M.G., An adaptation theory for nonparametric confidence intervals. Ann. Stat. 32 5 (2004) 1805–1840. Google Scholar
Cai, T. and Low, M.G., Adaptive confidence balls. Ann. Stat. 34 (2006) 202–228. Google Scholar
Chicken, E. and Cai, T., Block thresholding for density estimation: local and global adaptivity. J. Multivariate Anal. 95 (2005) 76–106. Google Scholar
I. Daubechies, Ten lectures on wavelets. SIAM Philadelphia (1992).
Donoho, D.L. and Johnstone, I.M., Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1996) 879–921. Google Scholar
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D., Density estimation by wavelet thresholding. Ann. Stat. 24 (1996) 508–539. Google Scholar
Dziedziul, K., Kucharska, M. and Wolnik, B., Estimation of the smoothness parameter. J. Nonparametric Stat. 23 (2011) 991–1001. Google Scholar
Giné, E. and Nickl, R., Confidence bands in density estimation. Ann. Stat. 38 (2010) 1122–1170. Google Scholar
Gloter, A. and Hoffmann, M., Nonparametric reconstruction of a multifractal function from noisy data. Probab. Theory Relat. Fields 146 (2010) 155187. Google Scholar
Hall, P. and Jones, M.C., Adaptive M-Estimation in Nonparametric Regression. Ann. Stat. 18 (1990) 1712–1728. Google Scholar
W. Härdle, G. Kerkyacharian, D. Picard and A.B. Tsybakov, Wavelets, Approximation and Statistical Applications. Springer-Verlag, New York (1998).
Hoffmann, M. and Nickl, R., On adaptive inference and confidence bands. Ann. Stat. 39 (2011) 2383–2409. Google Scholar
Horvath, L. and Kokoszka, P., Change-point detection with non parametric regression. Statistics: A J. Theoret. Appl. Stat. 36 (2002) 9–31. Google Scholar
Ingster, Y. and Stepanova, N., Estimation and detection of functions from anisotropic Sobolev classes. Electron. J. Stat. 5 (2011) 484–506. Google Scholar
Jaffard, S., Conjecture de Frisch et Parisi et généricité des fonctions multifractales. C. R. Acad. Sci. Paris Sér. I Math. 330 4 (2000) 265–270. Google Scholar
Low, M.G., On nonparametric confidence intervals. Ann. Stat. 25 (1997) 2547–2554. Google Scholar
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).
Ropela, S., Spline bases in Besov spaces. Bull. Acad. Pol. Sci. Serie Math. astr. Phys. 24 (1976) 319–325. Google Scholar
Sheather, S.J. and Jones, M.C., A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation. J. Royal Stat. Soc. Ser. B. 53 (1991) 683–690. Google Scholar