Wavelet estimators of a density constructed from observations of a mixture
Author:
D. Pokhyl’ko
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 70 (2005), 135-145
MSC (2000):
Primary 62G07; Secondary 42C40
DOI:
https://doi.org/10.1090/S0094-9000-05-00637-X
Published electronically:
August 12, 2005
MathSciNet review:
2109830
Full-text PDF Free Access
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Abstract: We construct projective estimators of a density by using a wavelet basis for the data being a sample from a mixture of several components whose concentrations vary with observations. We construct linear and adaptive estimators and prove that they converge in the mean square norm. We also prove that the linear estimator converges in the uniform norm.
References
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- Jérôme Kalifa and Stéphane Mallat, Thresholding estimators for linear inverse problems and deconvolutions, Ann. Statist. 31 (2003), no. 1, 58–109. MR 1962500, DOI https://doi.org/10.1214/aos/1046294458
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References
- R. E. Maĭboroda, Estimators of components of a mixture with varying concentrations, Ukrain. Mat. Zh. 48 (1996), no. 4, 562–566; English transl. in Ukrainian Math. J. 48 (1997) no. 4, 618–622. MR 1417019 (97j:62055)
- W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, Springer, New York, 1998. MR 1618204 (99f:42065)
- L. Devroye and L. Györfi, Nonparametric Density Estimation. The $L_1$-View, Wiley, New York, 1985. MR 0780746 (86i:62065)
- D. Donoho, I. Johnstone, G. Kerkyacharian, and D. Picard, Density estimation by wavelet tresholding, Ann. Statist. 24 (1996), 508–539. MR 1394974 (97f:62061)
- Ingrid Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1996. MR 1162107 (93e:42045)
- Jérôme Kalifa and Stéphane Mallat, Tresholding estimators on linear inverse problems and deconvolutions, Ann. Stat. 31 (2003), no. 1, 59–109. MR 1962500 (2003k:62229)
- R. Averkamp and C. Houdré, Wavelet tresholding for non-necessarily Gaussian noise: idealism, Ann. Stat. 31 (2003), no. 1, 110–151. MR 1962501 (2004a:62085)
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Additional Information
D. Pokhyl’ko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Mechanics and Mathematics Faculty, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email:
pokhid@ukr.net
Keywords:
Wavelets,
projective estimators of a density,
estimation of components in a mixture
Received by editor(s):
March 14, 2003
Published electronically:
August 12, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
