Theory of Probability and Mathematical Statistics
Lower bound for a dispersion matrix for the semiparametric estimation in a model of mixtures
O. V. Doronin
Abstract: The model of mixtures with varying concentrations is discussed. The parameterization of the first $ K$ of $ M$ components is considered. The semiparametric estimation technique based on the method of generalized estimating equations is considered. The consistency and asymptotic normality of estimators are proved. A lower bound for the dispersion matrix is found.
Keywords: Lower bound, mixture model, generalized estimating equations
1. A. A. Borovkov, Mathematical Statistics, Nauka'', Moscow, 1984; English transl., Gordon and Breach Science Publishers, Amsterdam, 1998. (Translated from the Russian by A. Moullagaliev and revised by the author)
2. 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)
3. 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.
4. R. E. Maiboroda and O. V. Sugakova, An estimation and classification by observations from a mixture, Kyiv University Press, Kyiv, 2008. (Ukrainian)
5. 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.
6. 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.
7. 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.
8. F. Autin and Ch. Pouet, Test on the components of mixture densities, Statistics & Risk Modelling 28 (2011), no. 4, 389-410.
9. 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.
10. P. Hall and X.-H. Zhou, Nonparametric estimation of component distributions in a multivariable mixture, Ann. Statist. 31 (2003), no. 1, 201-224.
11. R. E. Maiboroda and O. O. Kubaichuk, Improved estimators for moments constructed from observations of a mixture, Theor. Probability and Math. Statist. 70 (2005), 83-92.
12. R. Maiboroda and O. Sugakova, Nonparametric density estimation for symmetric distributions by contaminated data, Metrica 75 (2012), no. 1, 109-126.
13. 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.
14. 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.
15. G. J. McLachlan and D. Peel, Finite Mixture Models, Wiley, New York, 2000.
16. J. Shao, Mathematical statistics, Springer-Verlag, New York, 1998.
17. O. Sugakova, Adaptive estimates for the parameter of a mixture of two symmetric distributions, Theor. Probability and Math. Statist. 82 (2011), 149-159.
18. O. Sugakova, Empirical Bayesian classification for observations with admixture, Theor. Probability and Math. Statist. 84 (2012), 165-172.
19. D. M. Titterington, A. F. Smith, and U. E. Makov, Analysis of Finite Mixture Distributions, Wiley, New York, 1985.