Asymptotic normality of Kaplan–Meier estimators for mixtures with varying concentrations

Maĭboroda, R.

doi:10.1090/tpms/1039

Asymptotic normality of Kaplan–Meier estimators for mixtures with varying concentrations

Author: R. E. Maĭboroda
Translated by: N. N. Semenov
Journal: Theor. Probability and Math. Statist. 96 (2018), 133-144
MSC (2010): Primary 62N05, 62G05
DOI: https://doi.org/10.1090/tpms/1039
Published electronically: October 5, 2018
MathSciNet review: 3666877
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a modified Kaplan–Meier estimator for the distribution of components in a mixture with varying concentrations in the case of censored data. The asymptotic normality of this estimator is proved in the uniform norm.

References

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 https://doi.org/10.1016/j.jspi.2013.09.008
Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749
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 https://doi.org/10.1090/tpms/964
Ĭ. Ī. Gīhman and A. V. Skorohod, The theory of stochastic processes. I, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by S. Kotz; Die Grundlehren der mathematischen Wissenschaften, Band 210. MR 0346882
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 https://doi.org/10.1214/aos/1176347865
Jerald F. Lawless, Statistical models and methods for lifetime data, 2nd ed., Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2003. MR 1940115
Michel Loève, Probability theory. II, 4th ed., Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 46. MR 0651018
R. E. Maĭboroda, Estimation of the parameters of variable mixtures, Teor. Veroyatnost. i Mat. Statist. 44 (1991), 87–92 (Russian); English transl., Theory Probab. Math. Statist. 44 (1992), 85–88. MR 1168432
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, DOI https://doi.org/10.1007/BF02390622
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 https://doi.org/10.1080/10485252.2011.630076
R. Ē. Maĭboroda and V. G. Khīzanov, A modification of the Kaplan-Meier estimator for a model of a mixture with various concentrations, Teor. Ĭmovīr. Mat. Stat. 92 (2015), 103–109 (Ukrainian, with English and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 92 (2016), 109–116. MR 3619048, DOI https://doi.org/10.1090/tpms/986
A. Yu. Rizhov, Estimates for the distributions of components of a mixture based on censored data, Teor. Ĭmovīr. Mat. Stat. 69 (2003), 154–161 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 69 (2004), 167–174 (2005). MR 2110914, DOI https://doi.org/10.1090/S0094-9000-05-00623-X
Jun Shao, Mathematical statistics, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 2003. MR 2002723