A Dvoretzky-Kiefer-Wolfowitz type inequality for the Kaplan-Meier estimator
Bitouzé, D. ; Laurent, B. ; Massart, P.
Annales de l'I.H.P. Probabilités et statistiques, Tome 35 (1999), p. 735-763 / Harvested from Numdam
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@article{AIHPB_1999__35_6_735_0,
     author = {Bitouz\'e, D. and Laurent, B. and Massart, Pascal},
     title = {A Dvoretzky-Kiefer-Wolfowitz type inequality for the Kaplan-Meier estimator},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {35},
     year = {1999},
     pages = {735-763},
     mrnumber = {1725709},
     zbl = {1054.62589},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_1999__35_6_735_0}
}

Bitouzé, D.; Laurent, B.; Massart, P. A Dvoretzky-Kiefer-Wolfowitz type inequality for the Kaplan-Meier estimator. Annales de l'I.H.P. Probabilités et statistiques, Tome 35 (1999) pp. 735-763. http://gdmltest.u-ga.fr/item/AIHPB_1999__35_6_735_0/

[1] R.F. Bass, Law of iterated logarithm for set-indexed partial sum processes with finite variance, Z. Warscheinlichkeitstheor. Verw. Geb. 70 (1985) 591-608. | MR 807339 | Zbl 0575.60034

[2] L. Birgé and P. Massart, Rates of convergence for minimum contrast estimators, Probab. Theory Related Fields 97 (1993) 113-150. | MR 1240719 | Zbl 0805.62037

[3] L. Birgé and P. Massart, Minimum contrast estimators on sieves, Bernoulli 4 (3) (1998) 329-375. | MR 1653272 | Zbl 0954.62033

[4] D. Bitouzé, Estimation de fonctionnelles d'une densité à partir d'observations directes ou censurées, Ph.D. Thesis, Laboratoire de modélisation statistique et stochastique, Bât. 425, Université de Paris XI, 91405 Orsay Cédex, France, 1995 (in English).

[5] N. Breslow and J. Crowley, A large sample study of the life table and product limit estimators under random censorship, Ann. Statist. 11 (1974) 49-58. | MR 458674 | Zbl 0283.62023

[6] I.H. Dinwoodie, Large deviations for censored data, Ann. Statist. 21 (3) (1993) 1608-1620. | MR 1241281 | Zbl 0925.60020

[7] M.D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smimov theorems, Ann. Math. Statist. 23 (1952) 277-281. | MR 47288 | Zbl 0046.35103

[8] P. Doukhan, P. Massart and E. Rio, Invariance principles for absolutely regular empirical processes, Ann. Inst. Henri Poincaré 31 (2) (1995) 393-427. | Numdam | MR 1324814 | Zbl 0817.60028

[9] R.M. Dudley, Central limit theorems for empirical measures, Ann. Probab. 6 (6) (1978) 899-929. | MR 512411 | Zbl 0404.60016

[10] A. Dvoretzky, J.C. Kiefer and J. Wolfowitz, Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, Ann. Math. Statist. 33 (1956) 642-669. | MR 83864 | Zbl 0073.14603

[11] A. Földes and L. Rejtõ, A LIL type result for the product limit estimator, Z. Wahrscheinlichkeitstheor. Verw. Geb. 56 (1981) 75-86. | MR 612161 | Zbl 0443.62031

[12] R.D. Gill, Large sample behavior of the product limit estimator on the whole line, Ann. Statist. 11 ( 1983) 49-58. | MR 684862 | Zbl 0518.62039

[13] R.D. Gill, Glivenko-Cantelli for Kaplan-Meier, Math. Methods of Statistics 3 (1) (1994) 76-87. | MR 1272632 | Zbl 0824.62045

[14] R.D. Gill, Lectures on survival analysis, in: Lectures on Probability Theory, École d'Été de Probabilités de Saint-Flour XXII-1992, Lecture Notes in Mathematics, Vol. 1581, Springer, Berlin, 1994, pp. 115-241. | MR 1307414 | Zbl 0809.62028

[15] R.D. Gill and S. Johansen, A survey of product-integration with a view towards applications in survival analysis, Ann. Statist. 6 (1990) 1501-1555. | MR 1074422 | Zbl 0718.60087

[16] M.G. Gu and T.L. Lai, Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation, Ann. Probab. 18 (1) (1990) 160-189. | MR 1043942 | Zbl 0705.62040

[17] H.J. Hall and J.A. Wellner, Confidence bands for a survival curve from censored data, Biometrika 67 (1980) 133-143. | MR 570515 | Zbl 0423.62078

[18] E.L. Kaplan and P. Meier, Non-parametric estimation from incomplete observations, J. Amer. Statist. Assoc. 53 (1958) 562-563, 897-919. | MR 93867

[19] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality, Ann. Probab. 18 (3) (1990) 1269-1283. | MR 1062069 | Zbl 0713.62021

[20] M. Ossiander, A central limit theorem under metric entropy with L2 bracketing, Ann. Probab. 15 (1987) 897-919. | MR 893905 | Zbl 0665.60036

[21] G.R. Shorack and J.A. Wellner, Empirical Process with Applications to Statistics, Wiley, New York, 1986. | MR 838963 | Zbl 1170.62365

[22] W. Stute, Strong and weak representations of cumulative hazard function and Kaplan-Meier estimators on increasing sets, J. Statist. Planning and Inference 42 (1994) 315-329. | MR 1309627 | Zbl 0815.62016

[23] W. Stute and J.L. Wang, The strong law under random censorship, Ann. Statist. 21 (3) (1993) 1591-1607. | MR 1241280 | Zbl 0785.60020

[24] S. Van De Geer, Hellinger consistency of certain nonparametric maximum likelihood estimators, Ann. Statist. 21 (1) (1993) 14-44. | MR 1212164 | Zbl 0779.62033

[25] M.J. Van Der Laan, Efficient and inefficient estimation in semiparametric models, Ph.D. Thesis, Department of Mathematics, University of Utrecht, the Netherlands, 1993.

[26] M.J. Van Der Laan, Proving efficiency of NPMLE and identities, Technical Report 44, Group in Biostatistics, University of California, Berkeley, CA 94720, 1994.

[27] J.G. Wang, A note on the uniform consistency of the Kaplan-Meier estimator, Ann. Statist. 15 (1987) 1313-1316. | MR 902260 | Zbl 0631.62043