Volume 9, Issue 4, July 2020, Page: 154-161
On Survival Function Estimation in Dependent Partially Informative Random Censorship
Abdushukurov Abdurahim Ahmedovich, Department of Applied Mathematics and Informatics, Tashkent Branch of Moscow State University Named After M. V. Lamanosov, Tashkent, Uzbekistan
Bozorov Suxrob Baxodirovich, Faculty of Physics and Mathematics, Gulistan State University, Gulistan, Republic of Uzbekistan
Received: Jul. 8, 2020;       Accepted: Jul. 29, 2020;       Published: Aug. 10, 2020
DOI: 10.11648/j.ajtas.20200904.19      View  70      Downloads  26
In such areas as bio-medicine, engineering and insurance researchers are interested in positive variables, which are expressed as a time until a certain event. But observed data may be incomplete, because it is censored. Moreover, the random variables of interest (lifetimes) and censoring times can be influenced by other variable, often called prognostic factor or covariate. The basic problem is the estimation of survival function of lifetime. In this article we propose three asymptotical equivalent estimators of survival function in partially informative competing risks model. This paper deals with the estimation of a survival function with random right censoring and dependent censoring mechanism through covariate. We extend exponential – hazard, product - limit and relative - risk power estimators of survival functions in partially informative censoring model in which conditional on a covariate, the survival and censoring times are assumed to be independent. In this model, each observation is the minimum of one lifetime and two censoring times. The survival function of one of these censoring times is a power of the survival function of the lifetime. The distribution of the other censoring time has no relation with the distribution of the lifetime (non-informative censoring). For estimators we show their uniform strong consistency and convergence to same Gaussian process. Comparisons of estimators with the Jensen-Wiedmann’s estimator are included.
Random Censoring, Proportional Hazards Model, Exponential Hazard, Product-limit, Relative-risk, Survival Function
To cite this article
Abdushukurov Abdurahim Ahmedovich, Bozorov Suxrob Baxodirovich, On Survival Function Estimation in Dependent Partially Informative Random Censorship, American Journal of Theoretical and Applied Statistics. Vol. 9, No. 4, 2020, pp. 154-161. doi: 10.11648/j.ajtas.20200904.19
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A. A. Abdushukurov Nonparametric estimation of thedistributionfunction based on relative risk function. Commun Statist: Theory & Meth. 1998. V. 27. N. 8. p. 1991-2012.
A. A. Abdushukurov Onnonparametricestimationof reliability indices by censoredsample. Theory Probab. Appl. 1999. V. 43. N. 1. p. 3-11.
A. A. Abdushukurov, D. T. Nedzvedsky Asymptotic properties of empirical processes on censored samples of ramdomsices. J. Math. Sciences. 2005. V. 127. N. 1. p. 931-939.
A. A. Abdushukurov, D. Makhmudova Semiparametric estimation of distribution function in the informative model of competing risks. J. Math. Sciences. Springer. 2017. V. 227. N. 2, p. 117-123.
A. A. Abdushukurov Nonparametric estimation based on incomplete observations. In: International Encyclopedia of Statistical Sciences. (Prof. Miodrag Lovric, Editor). Springer. 2011. Pt. 14. p. 962-964.
A. A. AbdushukurovEstimates of unknown distributions from incomplete observations and its properties. LAMBERT Academic Publishing. 2011. p. 301. (Russian).
P. E. ChengNonparametric estimation of survival curve under dependentcensorship. J. Statist. Plann. Infer. 1989 V. 23. p. 181-191.
S. CsörgőEstimating in proportional hazards model of ran-dom censorship. // Statistics. 1988. v. 19. N. 3. p. 437-463.
U. Gather, J. Pawlitschko. Estimating the survival function under a generalized Koziol-Green model with partially informative censoring. Metrika. 1998. V. 48. p. 189-207.
T. Herbst Test of fit with the Koziol-Green model for random censership. Statist. Decisions. 1992. v. 10. p. 163-171.
U. Jensen, J. Wiedmann Estimation of a Survival Curve under Dependent Cenoring. Second Internat. Conf. on Math. Meth-s in Reliability. Bordeaux. France. July 4-7. 2000. V. 2. p. 571-574.
S. Kirmani, J. Y. Dauxois Testing the Koziol-Green model against monotone conditional odds for censoring. Statist. Probab. Lett. 2004. v. 63. N. 3. p. 327-334.
F. Siannis Applications of a parametric model for informative censoring. Biometrics. 2004. v. 60. N. 3. p. 704-714.
M. C. Wang, J. Qin, C. Chiang Analysing recurrent event data with informative censoring. J. Amer. Statist. Soc. 2001. v. 96. N. 455. p. 1057-1065.
H. Zhang, M. B. Rao On generalizedmaximumlikelihood estimation in the proportiona hazards model with partially informative censoring. Metrika. 2004. V. 59. p. 125-136.
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