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Csörgö, Sándor

Universal Gaussian approximations under random censorship. (English) Zbl 0868.62042

Ann. Stat. 24, No. 6, 2744-2778 (1996).
Summary: Universal Gaussian approximations are established for empirical cumulative hazard and product-limit processes under random censorship. They hold uniformly up to some large order statistics in the sample, with the approximation rates depending on the order of these statistics, and require no assumptions on the censoring mechanism. Weak convergence results and laws of the iterated logarithm follow on the whole line if the respective processes are stopped at certain large order statistics, depending on the type of result.
Some new consequences and negative results for confidence-band construction are discussed. Some new uniform consistency rates up to large order statistics are also derived and shown to be universally best possible for a wide range of tail order statistics.

Cited in 19 Documents

MSC:

62G20 Asymptotic properties of nonparametric inference
60F15 Strong limit theorems
62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles

Keywords:

universal Gaussian approximations; weak convergence; cumulative hazard; product-limit processes; random censorship; large order statistics; approximation rates; censoring; laws of the iterated logarithm; confidence-band construction; uniform consistency rates; tail order statistics
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[1] Aalen, O. O. (1976). Nonparametric inference in connection with multiple decrement models. Scand. J. Statist. 3 15-27. · Zbl 0331.62030
[2] Andersen, P. K., Borgan,., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York. · Zbl 0769.62061
[3] Bonvalot, F. and Castelle, N. (1991). Strong approximation of uniform empirical process by Kiefer process. Preprint 91-41, Univ. Paris-Sud, Orsay, France.
[4] 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. · Zbl 0283.62023 · doi:10.1214/aos/1176342705
[5] Bretagnolle, J. and Massart, P. (1989). Hungarian constructions from the nonasy mptotic viewpoint. Ann. Probab. 17 239-256. · Zbl 0667.60042 · doi:10.1214/aop/1176991506
[6] Burke, M. D., Cs örg o, S. and Horváth, L. (1981). Strong approximations of some biometric estimates under random censorship. Z. Wahrsch. Verw. Gebiete 56 87-112. · Zbl 0439.60012 · doi:10.1007/BF00531976
[7] Burke, M. D., Cs örg o, S. and Horváth, L. (1988). Correction to and improvement of ”Strong approximations of some biometric estimates under random censorship.” Probab. Theory Related Fields 79 51-57. · Zbl 0633.60054 · doi:10.1007/BF00319103
[8] Chen, K. and Ying, Z. (1996). A counterexample to a conjecture concerning the Hall-Wellner band. Ann. Statist. 24 641-646. · Zbl 0859.62046 · doi:10.1214/aos/1032894456
[9] Csáki, E. (1977). The law of the iterated logarithm for normalized empirical distribution function. Z. Wahrsch. Verw. Gebiete 38 147-167. · Zbl 0336.60029 · doi:10.1007/BF00533305
[10] Cs örg o, M., Cs örg o, S., Horváth, L. and Mason, D. M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14 31-85. · Zbl 0589.60029 · doi:10.1214/aop/1176992617
[11] Cs örg o, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York. · Zbl 0539.60029
[12] Cs örg o, S. (1988). Estimation in the proportional hazards model of random censorship. Statistics 19 437-463. · doi:10.1080/02331888808802115
[13] Cs örg o, S. (1989). Testing for the proportional hazards model of random censorship. In Proceedings of the Fourth Prague Sy mposium on Asy mptotic Statistics (1988) (P. Mandl and M. Hu sková, eds.) 41-53. Charles Univ. Press, Prague. · Zbl 0697.62040
[14] Cs örg o, S. and Horváth, L. (1982). Statistical inference from censored samples. Alkalmaz. Mat. Lapok 8 1-89. (In Hungarian.) · Zbl 0544.62044
[15] Cs örg o, S. and Horváth, L. (1983). The rate of strong uniform consistency for the product-limit estimator. Z. Wahrsch. Verw. Gebiete 62 411-426. · Zbl 0488.60043 · doi:10.1007/BF00535263
[16] Cs örg o, S. and Horváth, L. (1986). Confidence bands from censored samples. Canad. J. Statist. 14 131-144. JSTOR: · Zbl 0609.62069 · doi:10.2307/3314659
[17] Cs örg o, S. and Mielniczuk, J. (1988). Density estimation in the simple proportional hazards model. Statist. Probab. Lett. 6 419-426. · Zbl 0691.62039 · doi:10.1016/0167-7152(88)90102-2
[18] Efron, B. (1967). The two sample problem with censored data. Proc. Fifth Berkeley Sy mp. Math Statist. Probab. 4 831-853. Univ. California Press, Berkeley.
[19] Efron, B. (1981). Censored data and the bootstrap. J. Amer. Statist. Assoc. 76 312-319. · Zbl 0461.62039 · doi:10.2307/2287832
[20] Gill, R. D. (1980). Censoring and Stochastic Integrals. Math. Center Tracts 124. Math. Centrum, Amsterdam. · Zbl 0456.62003
[21] Gill, R. D. (1983). Large sample behavior of the product-limit estimator on the whole line. Ann. Statist. 11 49-58. · Zbl 0518.62039 · doi:10.1214/aos/1176346055
[22] Gill, R. D. (1994). Lectures on survival analysis. Ecole d’Eté de Probabilités de Saint-Flour XXII. Lecture Notes in Math. 1581 115-241. Springer, Berlin. · Zbl 0809.62028
[23] Gu, M. G. (1991). The Chung-Smirnov law for the product-limit estimator under random censorship. Chinese Ann. Math. Ser. B 12 96-105. · Zbl 0725.62046
[24] Gu, M. G. and Lai, T. L. (1990). Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation. Ann. Probab. 18 160-189. · Zbl 0705.62040 · doi:10.1214/aop/1176990943
[25] Hall, W. J. and Wellner, J. A. (1980). Confidence bands for a survival curve from censored data. Biometrika 67 133-143. JSTOR: · Zbl 0423.62078 · doi:10.1093/biomet/67.1.133
[26] Henze, N. (1993). A quick omnibus test for the proportional hazards model of random censorship. Statistics 24 253-263. · Zbl 0808.62045 · doi:10.1080/02331888308802412
[27] Kaplan, E. L. and Meier, P. (1958). Non-parametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457-481, 562-563. Koml ós, J., Major, P. and Tusnády, G. (1975a). An approximation of partial sums of independent rv’s and the sample df. Z. Wahrsch. Verw. Gebiete 32 111-131. Koml ós, J., Major, P. and Tusnády, G. (1975b). Weak convergence and embedding. In Limit Theorems of Probability Theory. Colloquia Mathematica Societatis János Boly ai (P. Révész, ed.) 11 149-165. North-Holland, Amsterdam. · Zbl 0089.14801
[28] Lo, S.-H. and Singh, K. (1985). The product-limit estimator and the bootstrap: some asy mptotic representations. Probab. Theory Related Fields 71 455-465. · Zbl 0561.62032 · doi:10.1007/BF01000216
[29] Major, P. and Rejt o, L. (1988). Strong embedding of the estimator of the distribution function under random censorship. Ann. Statist. 16 1113-1132. · Zbl 0667.62024 · doi:10.1214/aos/1176350949
[30] Mason, D. M. (1985). The asy mptotic distribution of generalized Rény i statistics. Acta Sci. Math. (Szeged) 48 315-323. · Zbl 0592.62016
[31] Mason, D. M. and van Zwet, W. R. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 871-884. · Zbl 0638.60040 · doi:10.1214/aop/1176992070
[32] Nair, V. N. (1984). Confidence bands for survival functions with censored data: a comparative study. Technometrics 26 265-275.
[33] Nelson, W. (1972). Theory and applications of hazard plotting for censored failure data. Technometrics 14 945-966.
[34] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[35] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. · Zbl 0538.62002
[36] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. Stute, W. (1994a). U-statistic processes: a martingale approach. Ann. Probab. 22 1725-1744. Stute, W. (1994b). Strong and weak representations of cumulative hazard functions and Kaplan- Meier estimators on increasing sets. J. Statist. Plann. Inference 42 315-329.
[37] Stute, W. and Wang, J.-L. (1993). The strong law under random censorship. Ann. Statist. 21 1591-1607. · Zbl 0785.60020 · doi:10.1214/aos/1176349273
[38] Wang, J.-G. (1987). A note on the uniform consistency of the Kaplan-Meier estimator. Ann. Statist. 15 1313-1316. · Zbl 0631.62043 · doi:10.1214/aos/1176350507
[39] Wellner, J. A. (1978). Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z. Wahrsch. Verw. Gebiete 45 73-88. · Zbl 0382.60031 · doi:10.1007/BF00635964
[40] Yang, S. (1992). Some inequalities about the Kaplan-Meier estimator. Ann. Statist. 20 535-544. · Zbl 0745.62037 · doi:10.1214/aos/1176348537
[41] Ying, Z. (1989). A note on the asy mptotic properties of the product-limit estimator on the whole line. Statist. Probab. Lett. 7 311-314. · Zbl 0675.62034 · doi:10.1016/0167-7152(89)90113-2
[42] Zhou, M. (1991). Some properties of the Kaplan-Meier estimator for independent nonidentically distributed random variables. Ann. Statist. 19 2266-2274. · Zbl 0756.62017 · doi:10.1214/aos/1176348399
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