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

DOI:
https://doi.org/10.1090/S0094-9000-07-00704-1

Published electronically:
July 5, 2007

MathSciNet review:
2336785

Full-text PDF

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.

**1.**L. Devroye and L. Gyorfi,*Nonparametric Density Estimation. The View*, John Wiley & Sons, Inc., New York, 1985. MR**780746 (86i:62065)****2.**Yu. V. Kozachenko,*Lectures on the Theory of Wavelets*, TBiMC, Kyiv, 2004. (Ukrainian)**3.**R. E. Maboroda,*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)****4.**R. E. Maboroda,*Statistical Analysis of Mixtures*, ``Kyiv University'', Kyiv, 2003. (Ukrainian)**5.**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)****6.**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****7.**I. Daubechies,*Ten Lectures on Wavelets*, SIAM, Philadelphia, 1996. MR**1162107 (93e:42045)****8.**D. Donoho, I. Johnstone, G. Kerkyacharian, and D. Picard,*Density estimation by wavelet thresholding*, Ann. Statist.**24**(1996), 508-539. MR**1394974 (97f:62061)****9.**W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov,*Wavelets, Approximation, and Statistical Applications*, Springer-Verlag, New York, 1998. MR**1618204 (99f:42065)****10.**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

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