Estimation of the parameters of the binomial distribution in a model of mixture

Author:
A. Shcherbina

Translated by:
S. V. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **86** (2012).

Journal:
Theor. Probability and Math. Statist. **86** (2013), 205-217

MSC (2010):
Primary 62F10; Secondary 62P10

Published electronically:
August 20, 2013

MathSciNet review:
2986460

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A model for observations sampled from a two component mixture is considered. Each object is associated with a certain numerical characteristic that may assume two values, namely zero (failure) or one (success) with identical probabilities for all objects in every class. Probabilities for these values are constant for all objects from the same component. The total numbers of objects of the first and second classes in groups as well as their characteristics are known. We study the problem of estimation of success probabilities for both components. We solve the problem by using the maximum likelihood method. We prove that the estimator is consistent and asymptotically normal. We apply the results obtained in the paper to a problem in genetics. An explicit form of the estimator and the asymptotic dispersion matrix is presented.

**1.**A. A. Borovkov,*Mathematical statistics*, Gordon and Breach Science Publishers, Amsterdam, 1998. Translated from the Russian by A. Moullagaliev and revised by the author. MR**1712750****2.**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**, 10.1007/BF02390622**3.**R. E. Maĭboroda,*Statistical Analysis of Mixtures*, Kyiv University Press, Kyiv, 2003. (Ukrainian).**4.**R. E. Maĭboroda and O. V. Sugakova,*An estimation and classification by observations from a mixture*, Kyiv University Press, Kyiv, 2008 (Ukrainian).**5.**A. M. Shscherbina,*Estimation of the mean value in a model of mixtures with varying concentrations*, Teor. Imovir. Matem. Statyst.**84**(2011), 142-154; English transl. in Theor. Probability and Math. Statist.**84**(2012), 151-164.**6.**A. M. Shscherbina,*A comparison of estimators of the mean values for mixtures with varying concentrations by using the generated data*, Visnyk Kyiv Nats. Univer. Ser. Mathematics. Mechanics**25**(2011), 43-47 (Ukrainian).**7.**O. Kubaychuk,*Estimation of moments by observations from mixtures with varying concentrations*, Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv, 2002), 2002, pp. 226–231. MR**2027394****8.**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****9.**Simon Newcomb,*A Generalized Theory of the Combination of Observations so as to Obtain the Best Result*, Amer. J. Math.**8**(1886), no. 4, 343–366. MR**1505430**, 10.2307/2369392**10.**K. Pearson,*Contribution to the mathematical theory of evolution*, Trans. Roy. Soc. A.**185**(1894), 71-110.**11.**A. Shcherbina and R. Maĭboroda,*Merging data from anonymous and open surveys: two-population problems*, Proceedings of the Baltic-Nordic-Ukrainian Summer School on Survey Statistics, ``TViMS'', Kyiv, 2009.**12.**D. M. Titterington, A. F. M. Smith, and U. E. Makov,*Statistical analysis of finite mixture distributions*, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1985. MR**838090**

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Additional Information

**A. Shcherbina**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine

Email:
artshcherbina@gmail.com

DOI:
https://doi.org/10.1090/S0094-9000-2013-00899-0

Keywords:
Estimation in a model of mixture,
parametric estimation,
genetic studies

Received by editor(s):
May 20, 2011

Published electronically:
August 20, 2013

Article copyright:
© Copyright 2013
American Mathematical Society