Browse > Article
http://dx.doi.org/10.4134/CKMS.c170130

DENSITY SMOOTHNESS PARAMETER ESTIMATION WITH SOME ADDITIVE NOISES  

Zhao, Junjian (Department of Mathematics College of Science Tianjin Polytechnic University)
Zhuang, Zhitao (College of Mathematics and Information Science North China University of Water Resources and Electric Power)
Publication Information
Communications of the Korean Mathematical Society / v.33, no.4, 2018 , pp. 1367-1376 More about this Journal
Abstract
In practice, the density function of a random variable X is always unknown. Even its smoothness parameter is unknown to us. In this paper, we will consider a density smoothness parameter estimation problem via wavelet theory. The smoothness parameter is defined in the sense of equivalent Besov norms. It is well-known that it is almost impossible to estimate this kind of parameter in general case. But it becomes possible when we add some conditions (to our proof, we can not remove them) to the density function. Besides, the density function contains impurities. It is covered by some additive noises, which is the key point we want to show in this paper.
Keywords
smoothness parameter estimation; density; wavelets; additive noise; Besov spaces;
Citations & Related Records
연도 인용수 순위
  • Reference
1 R. Li and Y. Liu, Wavelet optimal estimations for a density with some additive noises, Appl. Comput. Harmon. Anal. 36 (2014), no. 3, 416-433.   DOI
2 M. G. Low, On nonparametric confidence intervals, Ann. Statist. 25 (1997), no. 6, 2547-2554.   DOI
3 A. Martinez, Communication by energy modulation: the additive exponential noise channel, IEEE Trans. Inform. Theory 57 (2011), no. 6, 3333-3351.   DOI
4 S. J. Sheather and M. C. Jones, A reliable data-based bandwidth selection method for kernel density estimation, J. Roy. Statist. Soc. Ser. B 53 (1991), no. 3, 683-690.
5 J. Sun, H. Morrison, P. Harding, and M. Woodroofe, Density and mixture estimation from data with measurement errors, Technical report, 2002.
6 J. Zhao, The convergence of wavelet expansion with divergence-free properties in vector- valued Besov spaces, Appl. Math. Comput. 251 (2015), 143-153.
7 D. L. Donoho and I. M. Johnstone, Minimax estimation via wavelet shrinkage, Ann. Statist. 26 (1998), no. 3, 879-921.   DOI
8 T. T. Cai, Adaptive wavelet estimation: a block thresholding and oracle inequality approach, Ann. Statist. 27 (1999), no. 3, 898-924.   DOI
9 T. T. Cai and M. G. Low, Adaptive confidence balls, Ann. Statist. 34 (2006), no. 1, 202-228.   DOI
10 E. Chicken and T. T. Cai, Block thresholding for density estimation: local and global adaptivity, J. Multivariate Anal. 95 (2005), no. 1, 76-106.   DOI
11 D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996), no. 2, 508-539.   DOI
12 K. Dziedziul and B. Cmiel, Density smoothness estimation problem using a wavelet approach, ESAIM Probab. Stat. 18 (2014), 130-144.   DOI
13 J. Fan and J.-Y. Koo, Wavelet deconvolution, IEEE Trans. Inform. Theory 48 (2002), no. 3, 734-747.   DOI
14 A. Gloter and M. Hoffmann, Nonparametric reconstruction of a multifractal function from noisy data, Probab. Theory Related Fields 146 (2010), no. 1-2, 155-187.   DOI
15 W. Hardle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Lecture Notes in Statistics, 129, Springer-Verlag, New York, 1998.
16 L. Horvath and P. Kokoszka, Change-point detection with non-parametric regression, Statistics 36 (2002), no. 1, 9-31.   DOI
17 Y. Ingster and N. Stepanova, Estimation and detection of functions from anisotropic Sobolev classes, Electron. J. Stat. 5 (2011), 484-506.   DOI