A modified Kaplan–Meier estimator for a model of mixtures with varying concentrations
Authors:
R. E. Maĭboroda and V. G. Khizanov
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 92 (2016), 109-116
MSC (2010):
Primary 62N05, 62G05
DOI:
https://doi.org/10.1090/tpms/986
Published electronically:
August 10, 2016
MathSciNet review:
3619048
Full-text PDF Free Access
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Abstract: A modified Kaplan–Meier estimator for the distribution of components of a mixture with varying concentrations is constructed from censored data. The consistency of the estimators in the uniform norm is proved and the upper bound for the rate of convergence is derived.
References
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- Jun Shao, Mathematical statistics, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 2003. MR 2002723, DOI 10.1007/b97553
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- R. E. Maĭboroda and O. V. Sugakova, Estimation and classification by using observations sampled from a mixture, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- A. Yu. Ryzhov, Estimates of distributions of components in a mixture from censoring data, Teor. Ĭmovir. Mat. Stat. 69 (2003), 154–161; English transl. in Theor. Probability and Math. Statist. 69 (2004), 167–174.
- 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, Minimax rates over Besov spaces in ill-conditioned mixture-models with varying mixing-weights, J. Statist. Plann. Inference 146 (2014), 20–30. MR 3132476, DOI 10.1016/j.jspi.2013.09.008
- Richard D. Gill and Søren Johansen, A survey of product-integration with a view toward application in survival analysis, Ann. Statist. 18 (1990), no. 4, 1501–1555. MR 1074422, DOI 10.1214/aos/1176347865
References
- O. Korosteleva, Clinical Statistics: Introducing Clinical Trials, Survival Analysis, and Longitudinal Data Analysis, Jones and Bartlett Publishers, Sudbury, MA, 2008.
- C. Huber, N. Limnios, M. Mesbah, and M. Nikulin, Mathematical Methods in Survival Analysis, Reliability and Quality of Life, ISTE/Wiley, London/Hoboken, NJ, 2008. MR 2742532
- J. Shao, Mathematical Statistics, Springer-Verlag, New York, 1998. MR 2002723
- R. Maĭboroda and O. Sugakova, Statistics of mixtures with varying concentrations with application to DNA microarray data analysis, J. Nonparametric Statist. 24 (2012), no. 1, 201–215. MR 2885834
- R. E. Maĭboroda and O. V. Sugakova, Estimation and classification by using observations sampled from a mixture, “Kyiv University”, Kyiv, 2008. (Ukrainian)
- A. Yu. Ryzhov, Estimates of distributions of components in a mixture from censoring data, Teor. Ĭmovir. Mat. Stat. 69 (2003), 154–161; English transl. in Theor. Probability and Math. Statist. 69 (2004), 167–174.
- A. Shcherbina, Estimation of the mean value in a model of mixtures with varying concentrations, Teor. Ĭmovir. Mat. Stat. 84 (2011), 142–154; English transl. in Theor. Probability and Math. Statist. 84 (2012), 151–164. MR 2857425
- F. Autin and C. Pouet, Minimax rates over Besov spaces in ill-conditioned mixture-models with varying mixing-weights, J. Statist. Plann. Inference 146 (2014), 20–30. MR 3132476
- R. D. Gill and S. Johansen, A Survey of product-integration with a view toward application in survival analysis, Ann. Statist. 18 (1990), no. 4, 1501–1555. MR 1074422
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Additional Information
R. E. Maĭboroda
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
mre@univ.kiev.ua
V. G. Khizanov
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
vl.khizanov@gmail.com
Keywords:
Kaplan–Meier estimator,
models of mixtures with varying concentrations,
consistency,
censoring
Received by editor(s):
May 6, 2015
Published electronically:
August 10, 2016
Article copyright:
© Copyright 2016
American Mathematical Society
