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), 147162
MSC (2000):
Primary 62G07; Secondary 42C40
Published electronically:
July 5, 2007
MathSciNet review:
2336785
Fulltext PDF Free Access
Abstract 
References 
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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 thresholdbased estimator is considered for the integral metric. An adaptive modification of the thresholdbased estimator is constructed.
 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
(86i:62065)
 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
(97j:62055), http://dx.doi.org/10.1007/BF02390622
 4.
R. E. Maboroda, 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 (2005i:62068)
 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, CBMSNSF Regional
Conference Series in Applied Mathematics, vol. 61, Society for
Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
(93e:42045)
 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
(97f:62061), http://dx.doi.org/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,
SpringerVerlag, New York, 1998. MR 1618204
(99f:42065)
 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 WileyInterscience Publication. MR 1681904
(2000f:42023)
 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, 558562; English transl. in Ukrainian Math. J. 48 (1997), no. 4, 618622. 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), 121130; English transl. in Theory Probab. Math. Statist. 70 (2005), 135145. 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), 156166; English transl. in Theory Probab. Math. Statist. 59 (1999), 161171. 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), 508539. MR 1394974 (97f:62061)
 9.
 W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, SpringerVerlag, 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:
http://dx.doi.org/10.1090/S0094900007007041
PII:
S 00949000(07)007041
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
