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Density smoothness estimation problem using a wavelet approach

  • Karol Dziedziul (a1) and Bogdan Ćmiel (a2)


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.



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[1] Belitser, E. and Enikeeva, F., Empirical Bayesian Test of the Smoothness. Math. Methods Stat. 17 (2008) 118.
[2] Bull, A.D., A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets. Constructive Approximation 37 (2013) 295309.
[3] Bull, A.D., Honest adaptive confidence bands and self-similar functions. Electron. J. Stat. 6 (2012) 14901516.
[4] Cai, T., Adaptive Wavelet Estimation: A Block Thresholding and Oracle Inequality Approach. Ann. Stat. 27 (1999) 898924.
[5] Cai, T. and Low, M.G., An adaptation theory for nonparametric confidence intervals. Ann. Stat. 32 5 (2004) 18051840.
[6] Cai, T. and Low, M.G., Adaptive confidence balls. Ann. Stat. 34 (2006) 202228.
[7] Chicken, E. and Cai, T., Block thresholding for density estimation: local and global adaptivity. J. Multivariate Anal. 95 (2005) 76106.
[8] I. Daubechies, Ten lectures on wavelets. SIAM Philadelphia (1992).
[9] Donoho, D.L. and Johnstone, I.M., Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1996) 879921.
[10] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D., Density estimation by wavelet thresholding. Ann. Stat. 24 (1996) 508539.
[11] Dziedziul, K., Kucharska, M. and Wolnik, B., Estimation of the smoothness parameter. J. Nonparametric Stat. 23 (2011) 9911001.
[12] Giné, E. and Nickl, R., Confidence bands in density estimation. Ann. Stat. 38 (2010) 11221170.
[13] Gloter, A. and Hoffmann, M., Nonparametric reconstruction of a multifractal function from noisy data. Probab. Theory Relat. Fields 146 (2010) 155187.
[14] Hall, P. and Jones, M.C., Adaptive M-Estimation in Nonparametric Regression. Ann. Stat. 18 (1990) 17121728.
[15] W. Härdle, G. Kerkyacharian, D. Picard and A.B. Tsybakov, Wavelets, Approximation and Statistical Applications. Springer-Verlag, New York (1998).
[16] Hoffmann, M. and Nickl, R., On adaptive inference and confidence bands. Ann. Stat. 39 (2011) 23832409.
[17] Horvath, L. and Kokoszka, P., Change-point detection with non parametric regression. Statistics: A J. Theoret. Appl. Stat. 36 (2002) 931.
[18] Ingster, Y. and Stepanova, N., Estimation and detection of functions from anisotropic Sobolev classes. Electron. J. Stat. 5 (2011) 484506.
[19] Jaffard, S., Conjecture de Frisch et Parisi et généricité des fonctions multifractales. C. R. Acad. Sci. Paris Sér. I Math. 330 4 (2000) 265270.
[20] Low, M.G., On nonparametric confidence intervals. Ann. Stat. 25 (1997) 25472554.
[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).
[22] Ropela, S., Spline bases in Besov spaces. Bull. Acad. Pol. Sci. Serie Math. astr. Phys. 24 (1976) 319325.
[23] 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) 683690.


Density smoothness estimation problem using a wavelet approach

  • Karol Dziedziul (a1) and Bogdan Ćmiel (a2)


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