Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2017.03.28
  • Accepted : 2017.11.07
  • Published : 2018.10.31
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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.

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Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. T. T. Cai, Adaptive wavelet estimation: a block thresholding and oracle inequality approach, Ann. Statist. 27 (1999), no. 3, 898-924. https://doi.org/10.1214/aos/1018031262
  2. T. T. Cai and M. G. Low, Adaptive confidence balls, Ann. Statist. 34 (2006), no. 1, 202-228. https://doi.org/10.1214/009053606000000146
  3. E. Chicken and T. T. Cai, Block thresholding for density estimation: local and global adaptivity, J. Multivariate Anal. 95 (2005), no. 1, 76-106. https://doi.org/10.1016/j.jmva.2004.07.003
  4. D. L. Donoho and I. M. Johnstone, Minimax estimation via wavelet shrinkage, Ann. Statist. 26 (1998), no. 3, 879-921. https://doi.org/10.1214/aos/1024691081
  5. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996), no. 2, 508-539. https://doi.org/10.1214/aos/1032894451
  6. K. Dziedziul and B. Cmiel, Density smoothness estimation problem using a wavelet approach, ESAIM Probab. Stat. 18 (2014), 130-144. https://doi.org/10.1051/ps/2013030
  7. J. Fan and J.-Y. Koo, Wavelet deconvolution, IEEE Trans. Inform. Theory 48 (2002), no. 3, 734-747. https://doi.org/10.1109/18.986021
  8. 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. https://doi.org/10.1007/s00440-008-0187-1
  9. W. Hardle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Lecture Notes in Statistics, 129, Springer-Verlag, New York, 1998.
  10. L. Horvath and P. Kokoszka, Change-point detection with non-parametric regression, Statistics 36 (2002), no. 1, 9-31. https://doi.org/10.1080/02331880210930
  11. Y. Ingster and N. Stepanova, Estimation and detection of functions from anisotropic Sobolev classes, Electron. J. Stat. 5 (2011), 484-506. https://doi.org/10.1214/11-EJS615
  12. 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. https://doi.org/10.1016/j.acha.2013.07.002
  13. M. G. Low, On nonparametric confidence intervals, Ann. Statist. 25 (1997), no. 6, 2547-2554. https://doi.org/10.1214/aos/1030741084
  14. A. Martinez, Communication by energy modulation: the additive exponential noise channel, IEEE Trans. Inform. Theory 57 (2011), no. 6, 3333-3351. https://doi.org/10.1109/TIT.2011.2136850
  15. 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.
  16. J. Sun, H. Morrison, P. Harding, and M. Woodroofe, Density and mixture estimation from data with measurement errors, Technical report, 2002.
  17. J. Zhao, The convergence of wavelet expansion with divergence-free properties in vector- valued Besov spaces, Appl. Math. Comput. 251 (2015), 143-153.