An adaptive estimator of the density of components of a mixture
Author:
D. I. Pokhyl’ko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 74 (2007), 147-162
MSC (2000):
Primary 62G07; Secondary 42C40
DOI:
https://doi.org/10.1090/S0094-9000-07-00704-1
Published electronically:
July 5, 2007
MathSciNet review:
2336785
Full-text PDF Free Access
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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
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- Yu. V. Kozachenko, Lectures on the Theory of Wavelets, TBiMC, Kyiv, 2004. (Ukrainian)
- 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, DOI https://doi.org/10.1007/BF02390622
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- 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, DOI https://doi.org/10.1090/S0094-9000-05-00637-X
- 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
- 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
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References
- L. Devroye and L. Gyorfi, Nonparametric Density Estimation. The $L_1$ View, John Wiley & Sons, Inc., New York, 1985. MR 780746 (86i:62065)
- Yu. V. Kozachenko, Lectures on the Theory of Wavelets, TBiMC, Kyiv, 2004. (Ukrainian)
- R. E. Maĭboroda, Estimation of distributions of the components of mixtures having varying concentrations, Ukr. Matem. Zh. 48 (1996), no. 4, 558–562; English transl. in Ukrainian Math. J. 48 (1997), no. 4, 618–622. MR 1417019 (97j:62055)
- R. E. Maĭboroda, Statistical Analysis of Mixtures, “Kyiv University”, Kyiv, 2003. (Ukrainian)
- D. I. Pokhyl’ko, Wavelet estimators of the density constructed from observations of mixture, Teor. Imovir. Mat. Stat. 70 (2004), 121–130; English transl. in Theory Probab. Math. Statist. 70 (2005), 135–145. MR 2109830 (2005i:62068)
- O. V. Sugakova, Asymptotics of a kernel estimate for distribution density constructed from observations of a mixture with varying concentration, Teor. Imovir. Mat. Stat. 59 (1998), 156–166; English transl. in Theory Probab. Math. Statist. 59 (1999), 161–171. MR 1793776
- I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1996. MR 1162107 (93e:42045)
- D. Donoho, I. Johnstone, G. Kerkyacharian, and D. Picard, Density estimation by wavelet thresholding, Ann. Statist. 24 (1996), 508–539. MR 1394974 (97f:62061)
- W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Springer-Verlag, New York, 1998. MR 1618204 (99f:42065)
- B. Vidakovic, Statistical Modeling by Wavelets, John Wiley & Sons, New York, 1999. MR 1681904 (2000f:42023)
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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
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
