Adaptive estimation for a semiparametric model of mixture
Author:
O. V. Doronin
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 91 (2015), 29-41
MSC (2010):
Primary 62G05, 62G20, 62F12; Secondary 62P25, 62G30
DOI:
https://doi.org/10.1090/tpms/964
Published electronically:
February 3, 2016
MathSciNet review:
3364121
Full-text PDF Free Access
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Additional Information
Abstract: A model of mixture with varying concentrations is considered. It is assumed that the first $K$ of $M$, $1\le K\le M$, components of the mixture are parameterized. A technique of the adaptive semiparametric estimation is developed by using the generalized estimating equations. It is proved that the estimators are consistent and asymptotically normal.
References
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- A. Shcherbīna, Estimation of the mean in a model of a mixture with variable concentrations, Teor. Ĭmovīr. Mat. Stat. 84 (2011), 142–154 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 84 (2012), 151–164. MR 2857425, DOI https://doi.org/10.1090/S0094-9000-2012-00866-1
- A. Shcherbīna, Estimation of the parameters of the binomial distribution in a model of a mixture, Teor. Ĭmovīr. Mat. Stat. 86 (2011), 182–192 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 86 (2013), 205–217. MR 2986460, DOI https://doi.org/10.1090/S0094-9000-2013-00899-0
- Florent Autin and Christophe Pouet, Test on components of mixture densities, Stat. Risk Model. 28 (2011), no. 4, 389–410. MR 2877572, DOI https://doi.org/10.1524/strm.2011.1065
- Laurent Bordes, Céline Delmas, and Pierre Vandekerkhove, Semiparametric estimation of a two-component mixture model where one component is known, Scand. J. Statist. 33 (2006), no. 4, 733–752. MR 2300913, DOI https://doi.org/10.1111/j.1467-9469.2006.00515.x
- Peter Hall and Xiao-Hua Zhou, Nonparametric estimation of component distributions in a multivariate mixture, Ann. Statist. 31 (2003), no. 1, 201–224. MR 1962504, DOI https://doi.org/10.1214/aos/1046294462
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- Rostyslav Maiboroda and Olena Sugakova, Nonparametric density estimation for symmetric distributions by contaminated data, Metrika 75 (2012), no. 1, 109–126. MR 2878111, DOI https://doi.org/10.1007/s00184-010-0317-5
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- Geoffrey McLachlan and David Peel, Finite mixture models, Wiley Series in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2000. MR 1789474
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- O. Sugakova, Empirical-Bayesian classification for observations with an admixture, Teor. Ĭmovīr. Mat. Stat. 84 (2011), 155–162 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 84 (2012), 165–172. MR 2857426, DOI https://doi.org/10.1090/S0094-9000-2012-00858-2
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References
- A. A. Borovkov, Mathematical Statistics, “Nauka”, Moscow, 1984; English transl., Translated from the Russian by A. Moullagaliev and revised by the author, Gordon and Breach Science Publishers, Amsterdam, 1998. MR 1712750 (2000f:62003)
- O. V. Doronin, Lower bound of the dispersion matrix for the semiparametric estimation in a model of mixture, Teor. Imovirnost. Matem. Statist. (accepted)
- O. V. Doronin, Robust estimates for mixtures with a Gaussian component, Visnyk Kyiv National Taras Shevchenko University. Mathematics and Mechanics (2012), no. 1, 18–23. (Ukrainian)
- N. Lodatko and R. Maĭboroda, An adaptive moment estimator of a parameter of a distribution constructed from observations with admixture, Teor. Imovirnost. Matem. Statist. 75 (2006), 61–70; English transl in Theor. Probability and Math. Statist. 75 (2007), 71–82. MR 2321182 (2008g:62101)
- R. E. Maiboroda and O. V. Sugakova, An Estimation and Classification by Observations from a Mixture, Kyiv University Press, Kyiv, 2008. (Ukrainian)
- D. I. Pokhyl’ko, Wavelet estimators of a density constructed from observations of a mixture, Teor. Imovirnost. Matem. Statist. 70 (2004), 121–130; English transl in Theor. Probability and Math. Statist. 70 (2005), 135–145.
- A. M. Shcherbina, Estimation of the mean value in a model of mixtures with varying concentrations, Teor. Imovirnost. Matem. Statist. 84 (2011), 142–154; English transl in Theor. Probability and Math. Statist. 84 (2012), 151–164. MR 2857425 (2012h:62130)
- A. M. Shcherbina, Estimation of the parameters of the binomial distribution in a model of mixture, Teor. Imovirnost. Matem. Statist. 86 (2012), 182–192; English transl in Theor. Probability and Math. Statist. 86 (2013) 205–217. MR 2986460
- F. Autin and Ch. Pouet, Test on the components of mixture densities, Statistics & Risk Modelling 28 (2011), no. 4, 389–410. MR 2877572
- L. Bordes, C. Delmas, and P. Vandekerkhove, Semiparametric Estimation of a two-component mixture model where one component is known, Scand. J. Statist. 33 (2006), 733–752. MR 2300913 (2008f:62049)
- P. Hall and X.-H. Zhou, Nonparametric estimation of component distributions in a multivariable mixture, Ann. Statist. 31 (2003), no. 1, 201–224. MR 1962504 (2003m:62105)
- R. E. Maiboroda and O. O. Kubaichuk, Improved estimators for moments constructed from observations of a mixture, Teor. Imovir. Mat. Stat. 70 (2004), 74–81; English transl. in Theor. Probability and Math. Statist. 70 (2005), 83–92.
- R. Maiboroda and O. Sugakova, Nonparametric density estimation for symmetric distributions by contaminated data, Metrica 75 (2012), no. 1, 109–126. MR 2878111
- R. Maiboroda and O. Sugakova, Statistics of mixtures with varying concentrations with application to DNA microarray data analysis, J. Nonparam. Statist. 24 (2012), no. 1, 201–205. MR 2885834
- R. E. Maiboroda, O. V. Sugakova, and A. V. Doronin, Generalized estimating equations for mixtures with varying concentrations, Canadian J. Statist. 41 (2013), no. 2, 217–236. MR 3061876
- G. J. McLachlan and D. Peel, Finite Mixture Models, Wiley, New York, 2000. MR 1789474 (2002b:62025)
- J. Shao, Mathematical Statistics, Springer-Verlag, New York, 1998. MR 2002723 (2004g:62002)
- O. Sugakova, Adaptive estimates for the parameter of a mixture of two symmetric distributions, Teor. Imovir. Mat. Stat. 82 (2010), 146–155 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theor. Probability and Math. Statist. 82 (2011), 149–159. MR 2790490 (2011m:62130)
- O. Sugakova, Empirical Bayesian classification for observations with an admixture, Teor. Imovir. Mat. Stat. 84 (2011), 155–162 (Ukrainian, with English, Russian and Ukrainian summaries); English transl. Theor. Probability and Math. Statist. 84 (2012), 165–172. MR 2857426 (2012f:62128)
- D. M. Titterington, A. F. Smith, and U. E. Makov, Analysis of Finite Mixture Distributions, Wiley, New York, 1985.
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Additional Information
O. V. Doronin
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email:
al_doronin@ukr.net
Keywords:
Adaptive estimation,
model of mixture,
generalized estimating equations
Received by editor(s):
March 4, 2014
Published electronically:
February 3, 2016
Article copyright:
© Copyright 2016
American Mathematical Society