Adaptive test on means homogeneity by observations from a mixture
Authors:
R. E. Maĭboroda and O. V. Sugakova
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 93 (2016), 123-135
DOI:
https://doi.org/10.1090/tpms/998
Published electronically:
February 7, 2017
MathSciNet review:
3553445
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: We consider the problem of testing the homogeneity of two components of a mixture with varying mixing probabilities and construct an adaptive test that minimizes the asymptotic probability of error of the second kind for local alternatives.
References
- A. A. Borovkov, Probability theory, Gordon and Breach Science Publishers, Amsterdam, 1998. Translated from the 1986 Russian original by O. Borovkova and revised by the author. MR 1711261
- R. Maĭboroda and O. Kubaĭchuk, Improved estimates for moments based on observations from a mixture, Teor. Ĭmovīr. Mat. Stat. 70 (2004), 74–81 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 70 (2005), 83–92. MR 2109826, DOI 10.1090/S0094-9000-05-00642-3
- O. V. Doronīn, Adaptive estimation in a semiparametric mixture model, Teor. Ĭmovīr. Mat. Stat. 91 (2014), 26–37 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 91 (2015), 29–41. MR 3364121, DOI 10.1090/tpms/964
- R. E. Maĭboroda and O. V. Sugakova, Estimation and Classification by Observations of a Mixture, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- A. Yu. Ryzhov, A test of the hypothesis about the homogeneity of components of a mixture with varying concentrations by using censored data, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 72 (2005), 129–139; English transl. in Theory Probab. Math. Statist. 72 (2006) 145–155.
- 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 10.1090/S0094-9000-2012-00866-1
- Florent Autin and Christophe Pouet, Test on components of mixture densities, Stat. Risk Model. 28 (2011), no. 4, 389–410. MR 2877572, DOI 10.1524/strm.2011.1065
- Alexey Doronin and Rostyslav Maiboroda, Testing hypotheses on moments by observations from a mixture with varying concentrations, Mod. Stoch. Theory Appl. 1 (2014), no. 2, 195–209. MR 3316487, DOI 10.15559/15-VMSTA19
- 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, DOI 10.1002/0471721182
- Rostyslav Maiboroda and Olena Sugakova, Statistics of mixtures with varying concentrations with application to DNA microarray data analysis, J. Nonparametr. Stat. 24 (2012), no. 1, 201–215. MR 2885834, DOI 10.1080/10485252.2011.630076
- 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
References
- A. A. Borovkov, Probability Theory, “Nauka”, Moscow, 1986; English transl., Gordon and Breach Science Publishers, Amsterdam, 1998. MR 1711261
- O. O. Kubaĭchuk and R. E. Maĭboroda, Improved estimates for moments based on observations from a mixture, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 70 (2004), 74–81; English transl. in Theory Probab. Math. Statist. 70 (2005), 83–92. MR 2109826
- O. V. Doronin, Adaptive estimation in a semiparametric mixture model, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 91 (2014), 26–37; English transl. in Theory Probab. Math. Statist. 91 (2015), 29–41. MR 3364121
- R. E. Maĭboroda and O. V. Sugakova, Estimation and Classification by Observations of a Mixture, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- A. Yu. Ryzhov, A test of the hypothesis about the homogeneity of components of a mixture with varying concentrations by using censored data, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 72 (2005), 129–139; English transl. in Theory Probab. Math. Statist. 72 (2006) 145–155.
- A. Shcherbina, Estimation of the mean value in a model of mixtures with varying concentrations, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 84 (2011), 142–154; English transl. in Theory Probab. Math. Statist 84 (2012) 151–164. MR 2857425
- F. Autin and Ch. Pouet, Test on the components of mixture densities, Stat. Risk Model. 28 (2011), no. 4, 389–410. MR 2877572
- A. Doronin and R. Maiboroda, Testing hypotheses on moments by observations from a mixture with varying concentrations, Mod. Stoch. Theory Appl. 1 (2014), no. 2, 195–209. MR 3316487
- G. J. McLachlan and D. Peel, Finite Mixture Models, Wiley-Interscience, New York, 2000. MR 1789474
- R. Maiboroda and O. Sugakova, Statistics of mixtures with varying concentrations with application to DNA microarray data analysis, J. Nonparametr. Stat. 24 (2012), no. 1, 201–215. MR 2885834
- D. M. Titterington, A. F. M. Smith, and U. E. Makov, Statistical Analysis of Finite Mixture Distribution, Wiley, New York, 1985. MR 838090
Additional Information
R. E. Maĭboroda
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
mre@univ.kiev.ua
O. V. Sugakova
Affiliation:
Department of Higher Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
sugak@univ.kiev.ua
Keywords:
Adaptive algorithms,
local alternatives,
models of mixtures with varying concentrations,
test for homogeneity of two means
Published electronically:
February 7, 2017
Additional Notes:
This paper was prepared following the talk at the International conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright:
© Copyright 2017
American Mathematical Society
