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Estimation of a Distribution Function from Incomplete Observations

Published online by Cambridge University Press:  05 September 2017

Paul Meier
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Abstract

The product-limit estimator for a distribution function, appropriate to observations which are variably censored, was introduced by Kaplan and Meier in 1958; it has provided a basis for study of more complex problems by Cox and by others. Its properties in the case of random censoring have been studied by Efron and later writers. The basic properties of the product-limit estimator are here shown to be closely parallel to the properties of the empirical distribution function in the general case of variably and arbitrarily censored observations.

Type
Part III — Statistical Theory
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 67 - 87
Copyright
Copyright © 1975 Applied Probability Trust 

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References

[1] Aalen, O. (1973) Nonparametric inference in connection with multiple decrement models. Statistical Research Report (mimeo) , Institute of Mathematics, University of Oslo.Google Scholar
[2] Bartlett, M. S. (1966) An Introduction to Stochastic Processes with Special Reference to Methods and Applications. University Press, Cambridge.Google Scholar
[3] Berkson, J. and Gage, R. P. (1950) Calculation of survival rates for cancer. Proc. Mayo Clinic 25, 270–286.Google ScholarPubMed
[4] Billingsley, P. (1968) Weak Convergence of Probability Measures. Wiley, New York.Google Scholar
[5] Böhmer, P. E. (1912) Theorie der unabhängigen Wahrscheinlichkeiten. Rapports, Mémoires et Procès-verbaux du Septième Congrès International d'Actuaires, Amsterdam. 2, 327–343.Google Scholar
[6] Breslow, N. and Crowley, J. (1974) A large sample study of the life table and product limit estimates under random censorship. Ann. Statist. 2, 437–453.Google Scholar
[7] Cox, D. R. (1972) Regression models and life tables. J. R. Statist. Soc. Ser. B 34, 187–220.Google Scholar
[8] Donsker, M. (1952) Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23, 277–281.Google Scholar
[9] Doob, J. L. (1949) Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 20, 393–403.Google Scholar
[10] Efron, B. (1967) The two sample problem with censored data. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 4, 831–853.Google Scholar
[11] Greenwood, M. (1926) The natural duration of cancer. Reports on Public Health and Medical Subjects 33, H. M. Stationary Office, London.Google Scholar
[12] Irwin, J. O. (1949) The standard error of an estimate of expectation of life. J. Hygiene 47, 188–189.Google Scholar
[13] Kaplan, E. L. and Meier, P. (1958) Non-parametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53, 457–481.Google Scholar
[14] Meier, P. (1967) Nonparametric estimation from incomplete observations — II. (Unpublished MS.) Google Scholar
[15] Peto, R. and Peto, J. (1972) Asymptotically efficient rank invariant test procedures. J. R. Statist. Soc. Ser. A 135, 185–198.Google Scholar
[16] Stephan, F. F. (1945) The expected value and variance of the reciprocal and other negative powers of a positive Bernoullian variate. Ann. Math. Statist. 16, 50–61.Google Scholar
[17] Wagner, S. S. and Altman, S. A. (1973) What time do the baboons come down from the trees? (An estimation problem.) Biometrics 29, 623–635.CrossRefGoogle Scholar