Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 

 

An adaptive estimator of the density of components of a mixture


Author: D. I. Pokhyl'ko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 74 (2006).
Journal: Theor. Probability and Math. Statist. 74 (2007), 147-162
MSC (2000): Primary 62G07; Secondary 42C40
Published electronically: July 5, 2007
MathSciNet review: 2336785
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Linear and nonlinear wavelet estimators of the density of components of a mixture are considered in the paper. The rate of convergence in the uniform metric and large deviation probabilities are obtained for linear estimators. The limit behavior of the threshold-based estimator is considered for the integral metric. An adaptive modification of the threshold-based estimator is constructed.


References [Enhancements On Off] (What's this?)

  • 1. Luc Devroye and László Györfi, Nonparametric density estimation, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics, John Wiley & Sons, Inc., New York, 1985. The 𝐿₁ view. MR 780746
  • 2. Yu. V. Kozachenko, Lectures on the Theory of Wavelets, TBiMC, Kyiv, 2004. (Ukrainian)
  • 3. R. Ē. Maĭboroda, Estimation of the distributions of the components of mixtures having varying concentrations, Ukraïn. Mat. Zh. 48 (1996), no. 4, 558–562 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 4, 618–622 (1997). MR 1417019, 10.1007/BF02390622
  • 4. R. E. Ma{\u{\i\/}}\kern.15emboroda, Statistical Analysis of Mixtures, ``Kyiv University'', Kyiv, 2003. (Ukrainian)
  • 5. D. Pokhil′ko, Wavelet estimates for density from observations of a mixture, Teor. Ĭmovīr. Mat. Stat. 70 (2004), 121–130 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 70 (2005), 135–145. MR 2109830, 10.1090/S0094-9000-05-00637-X
  • 6. O. V. Sugakova, Asymptotics of a kernel estimate for the density of a distribution constructed from observations of a mixture with varying concentration, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 156–166 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 161–171 (2000). MR 1793776
  • 7. Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
  • 8. David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996), no. 2, 508–539. MR 1394974, 10.1214/aos/1032894451
  • 9. Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, and Alexander Tsybakov, Wavelets, approximation, and statistical applications, Lecture Notes in Statistics, vol. 129, Springer-Verlag, New York, 1998. MR 1618204
  • 10. Brani Vidakovic, Statistical modeling by wavelets, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1681904

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 62G07, 42C40

Retrieve articles in all journals with MSC (2000): 62G07, 42C40


Additional Information

D. I. Pokhyl'ko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01033, Ukraine
Email: pdi_2004@mail.ru

DOI: https://doi.org/10.1090/S0094-9000-07-00704-1
Keywords: Wavelets, mixture, estimator of the density, adaptive estimator, projective estimator
Received by editor(s): June 27, 2005
Published electronically: July 5, 2007
Article copyright: © Copyright 2007 American Mathematical Society