Editorial Manager(tm) for Lifetime Data Analysis Manuscript Draft. Manuscript Number: Title: On An Exponential Bound for the Kaplan - Meier Estimator

Transcription

1 Editorial Managertm) for Lifetime Data Analysis Manuscript Draft Manuscript Number: Title: On An Exponential Bound for the Kaplan - Meier Estimator Article Type: SI-honor of N. Breslow invitation only) Section/Category: Keywords: random censoring; right censoring; product limit estimator; inequalities; universal limit theorems; uniform limit theorems; confidence bands; empirical distribution Corresponding Author: Prof. J. A Wellner, Corresponding Author's Institution: University Of Washington First Author: J. A Wellner Order of Authors: J. A Wellner Manuscript Region of Origin: Abstract:

2 * Title Page w/ ALL Author Contact Info. On An Exponential Bound for the Kaplan - Meier Estimator Jon A. Wellner University of Washington Abstract: We review limit theory and inequalities for the Kaplan-Meier product limit estimator of a survival function on the whole line R. Along the way we provide bounds for the constant in an interesting inequality due to Bitouzé, Laurent, and Massart, and provide some numerical evidence in support of one of their conectures. Jon A. Wellner University of Washington Department of Statistics Box Seattle, WA - aw@stat.washington.edu Supported in part by NSF grant DMS-0 and by NI-AID grant R0 AI- 0 AMS 000 subect classifications: Primary G, G0; secondary G0 Keywords and phrases: random censoring, right censoring, product limit estimator, inequalities, universal limit theorems, uniform limit theorems, confidence bands, empirical distribution

3 * Title Page w/ ALL Author Contact Info On An Exponential Bound for the Kaplan - Meier Estimator Jon A. Wellner University of Washington Abstract: We review limit theory and inequalities for the Kaplan-Meier Kaplan and Meier ) product limit estimator of a survival function on the whole line R. Along the way we provide bounds for the constant in an interesting inequality due to Bitouzé, Laurent and Massart ), and provide some numerical evidence in support of one of their conectures.. Introduction: the Kaplan-Meier estimator Suppose that X,...,X n,... are independent and identically distributed with distribution function F, and suppose that Y,...,Y n are independent and identically distributed with distribution function G. Suppose that we only observe W i = Z i,δ i ) where Z i = X i Y i, δ i = Xi Y i ]. Typically in survival analysis the X i s represent lifetimes and the Y i s represent censoring times. The above model for right-)censored data is sometimes called the random right censoring model. Define the empirical sub-distribution functions of the uncensored and censored Z s by and define Λ n by Λ n x) = H uc n z) = n H c n z) = n n δ i Zi z], i= n δ i ) Zi z], i= H n z) = H uc n z) + H c nz),,x] H n z ) dhuc n z). Supported in part by NSF grant DMS-0 and by NI-AID grant R0 AI- 0 AMS 000 subect classifications: Primary G, G0; secondary G0 Keywords and phrases: random censoring, right censoring, product limit estimator, inequalities, universal limit theorems, uniform limit theorems, confidence bands, empirical distribution

4 0 0 0 Wellner/Exponential Bound for Kaplan - Meier Then the Kaplan-Meier or product limit) estimator F n of the distribution function F is defined by F n x) = z x Λ n z)) = z:z i x ) n δi Ri n R i + where R i is the rank of Z i, δ i ) in the set {Z, δ ), =,...,n} with lexicographical ordering. For the empirical distribution function F n of the X i s, we know from Dvoretzky, Kiefer and Wolfowitz ) and Massart 0) that P n F n F > λ) exp λ ) ) for all λ > 0. In the spirit of the exponential bound ), Bitouzé, Laurent and Massart ) established the following exponential bound for the supremum distance between F n and F weighted by G: Theorem. Let F n be the Kaplan-Meier estimator of the distribution function F. There is an absolute constant C such that, for any λ > 0, P n G) F n F) > λ).exp λ + Cλ). ) As noted by Bitouzé, Laurent and Massart ), Of course, it would be desirable to compute C, but unfortunately, our techniques are not sharp enough to do that efficiently. One goal of this paper is to provide the following bound for the constant C: Proposition. The constant C in ) satisfies C. In the course of the proof of this proposition, we also provide explicit bounds on the constants involved in the bracketing entropy for bounded monotone functions given in Theorem.. of van der Vaart and Wellner ); see Theorem below. This bound on the numerical constants in that theorem may be of independent interest. Bitouzé, Laurent and Massart ) also say page ) that: We do not know whether the inequality P n G) b F n F) > λ) exp λ ) ) holds or not for the Kaplan-Meier estimator. Indeed, this is a natural question since on the one hand we know... that it holds asymptotically, and on the other hand that it is valid for all n in the non censored case see Massart 0)). Inequality ) does not of course imply ), but can be considered as a first step toward such a result.

5 0 0 0 Wellner/Exponential Bound for Kaplan - Meier Of course the bound provided by Proposition is absurdly large, especially in view of the numerical evidence in favor of ), but it gives a strong indication of the limitations of the current method of proof. Significant improvements of the current bound will likely involve the use of local entropy as discussed in van de Geer ) and van de Geer 000), or improvements of the bracketing entropy bound obtained here in Theorem, or both. Here is some numerical evidence in support of ). The following two scenarios and plots have been selected from a list of about thirty different scenarios. The first scenario is F = exponential), G = Uniform0, ], n = 00, and m = 00 Monte-Carlo replications. The dashed line shows the conectured bound exp λ ) ), while the solid line is the empirical survival function or one minus the empirical distribution function) for the m = 00 computed values of n G) F n F). In the second scenario, F = exponential), Gt) = t/).0, n = 00, and m = 00. The dashed line shows the conectured bound exp λ ) ), while the solid line is the empirical survival function or one minus the empirical distribution function) for the m = 00 computed values of n G) F n F) Fig. Empirical survival function solid) and conectured bound dashed) n = 00, F = exponential), G = Uniform0, )

6 Wellner/Exponential Bound for Kaplan - Meier 0... Fig. Empirical survival function solid) and conectured bound dashed) n = 00, F = exponential), Gt) = t/).0. Bracketing entropy for monotone functions Let F denote a class of uniformly bounded monotone functions on the real line; without loss we assume that f : R 0,] for all f F. Theorem. The class F of monotone functions f : R 0,] satisfies log N ], F,L r Q)) K r for every probability measure Q, every r, and a constant K r = C r r r + C r )) /r logl r) where C r = / /) /) ) and L r = /r + ). In particular C. and K.. Proof. This is proved in Section.. of van der Vaart and Wellner ), pages -. The constant follows from that proof by careful bookkeeping; see Section for details.. Limit theory for the Kaplan - Meier estimator In this section we give a very brief review of known limit theory for the Kaplan - Meier estimator F n, with emphasis on results that hold on either

7 0 0 0 Wellner/Exponential Bound for Kaplan - Meier all R or large subintervals of 0,τ H ] where, for any distribution function F, let τ F inf{t : Ft) = }. Donsker theorems: The original weak convergence theorem for the Kaplan-Meier estimator established by Breslow and Crowley ) showed that n Fn F) F)BC) in D0,T] ) for fixed T < τ H where B is a standard Brownian motion process on 0, ) and Ct) H ) dhuc = 0,t] F ) G ) df. 0,t] It was noted by Hall and Wellner 0) that ) could be rephrased as n K F F n F) B 0 K) in D0,T] ) where B 0 is a standard Brownian bridge process on 0,] and K C/+C). The weak convergence result in ) was improved to a strong approximation result by Burke, Csörgő and Horváth ) as follows: consider intervals of the form 0,T n ] with T n ր τ H satisfying HT n ) δ log n/n) / where δ > 0. Then F n can be defined on a common probability space with a sequence of) Brownian motion processes) B n so that Pr sup 0 t T n n F n t) Ft)) Ft))B n t) r n ) Cn δ where C is a constant, b n / HT n )), and r n = Omax{n / b nlog n) /, n / b n log n, n / b nlog n) }). In a different direction, Gill ) showed that { } n K F F Zn) n F) B 0 K) in D0,τ H ] ) where Z n) max i n Z i and for any process W the stopped process W T W T). Moreover, Gill ) showed that if then τh 0 df G <, ) { n } Zn) Fn F) F)BC) in D0,τH ] )

8 0 0 0 and Wellner/Exponential Bound for Kaplan - Meier { n K } Zn) F F n F) B 0 K) in D0,τ H ]. ) These conclusions were strengthened by Ying ) who showed that stopping of the process at Z n) is not needed; e.g. and if ) holds, then and n K F F n F) B 0 K) in D0,τ H ], ) n Fn F) F)BC) in D0,τ H ] ) K n F F n F) B 0 K) in D0,τ H ]. ) On the other hand Chen and Ying ) showed that ) can fail if ) does not hold, providing a counterexample to a conecture of Hall and Wellner 0). See Gillespie and Fisher ), Nair ), Csörgő and Horváth ), Hollander and Peña ), Hollander, McKeague and Yang ), and McKeague and Zhao 00) for other related work on confidence bands. Glivenko-Cantelli theorems: After incorrect proofs of Glivenko-Cantelli type theorems for the Kaplan-Meier estimator by Gill 0) and Shorack and Wellner ), Wang ) succeeded in establishing the first correct and clean) theorem. Wang showed that always hold, and hence that sup t<τ H Ft) Ft) p 0, and ) sup t<z n) Ft) Ft) p 0 ) sup Ft) Ft) p 0 ) t τ H if and only if Gτ H ) < or Fτ H ) = F {τ H } = 0. Wang ) also pointed out an error in Shorack and Wellner ) and gave a correct statement of the result which could be proved by the methods used by Shorack and Wellner. Stute and Wang ) showed that ) also holds with a.s. under the same conditions. Gill ) provided an alternative proof of the result of Stute and Wang ). Stute a) obtained a

9 0 0 0 Wellner/Exponential Bound for Kaplan - Meier weighted Glivenko-Cantelli type theorem by showing that if ψ = ψ + ψ is a U type weight function where ψ is and ψ is, then if it follows that τh 0 ψdf <, df <, and G τh ψ 0 sup ψt) F n t) Ft) p 0. t τ H df <, C / df <, Stute a) also indicated how this could be extended to an almost sure result. Univeral approximation theorems on increasing sets: Gu and Lai 0) and Gu ) apparently initiated the study of sup t Z n kn) Λ n t) Λt) sup F n t) Ft) t Z n kn) Ft) = O ) for k n {,...,n }. Stute b) continued this study and showed that if n k n )/n and k n /log n, then for each > 0 log n) + almost surely, and k n + sup t Z n kn)) F n t) Ft) = Oγ n ). sup t Z n kn) Λ n t) Λt) = log n)+)/ kn ) Oγ n ) The results of Stute b) were improved and sharpened by Csörgő ) who showed that log ) O n kn almost surely ) O p kn in probability. ) and similarly with the left side in ) replaced by the supremum in ). The almost sure part of Csörgő s bounds were further improved by Giné and Guillou ) who showed that the log n in the almost sure bounds can be replaced by log log n. Alternative derivations of limit theory for the Kaplan - Meier estimator F n under the assumption of constant censoring times have been given by Meier ) and Pollard 0). It would be interesting to try to extend the results of these authors to the whole line or 0,τ H ] with some appropriate re-definition of 0,τ H ]. For a treatment of some rate of convergence results in the encompassing framework of variable censoring in which each censoring time Y i is allowed to have its own distribution G i, see Csörgő and Horváth ), section, pages -.

10 Proof of Proposition Wellner/Exponential Bound for Kaplan - Meier From Theorem of Bitouzé, Laurent and Massart ), the proof on page, and rephrasing in the notation of van der Vaart and Wellner ), it follows that ) k P sup sup n G λ kf) > λ + c 0 σ + c ϕ 0 ) k n f F.e H 0) exp λ /a ) ) where c =./, c =., 0 < 0 <, sup f F V ar P f) σ a /, and t t ϕt) = log N ] x, F,L P)) dx Hx) dx. 0 In the context of Bitouzé, Laurent and Massart ), Corollary, ϕt) t 0 γ/xdx = γt / where γ K. by Theorem, so the inequality in ) becomes ) k P sup sup n G kf) > λ + c 0 ) + c 0 k n f F where c = γc = K c =. K. Set ξ λ + c 0 ) + c 0 ; then for ξ c 0, P sup sup k n f F Note that for 0 < /c we have 0.e γ/ 0 exp λ /a ) ) ) k n G kf) > ξ.e γ/ 0 exp a ξ c 0 + c 0 ) ξ c 0 ξ c 0 ) c 0 ) + c 0 ) ) ξ c 0 ξ c 0 ξ c 0 ) ξ c 0 ξ c 0 ξ.. 0) Taking 0 = b/ξ /c ) for a constant b which will be chosen appropri-

11 0 0 0 Wellner/Exponential Bound for Kaplan - Meier ately) so that ξ bc, the bound becomes γξ. exp b + c ξ a b + c ) bξ a exp ξ /a ) ) =.exp a γc ) / ξ + c γ / ξ a c /a ) / exp ξ /a ) by choosing b = a γ/c ) so that ξ ac γ/c ) = a γc, and hence ξ aγc ) /,.exp a γc ) / ξ + c c /a ) c c /a ) / γ/ ξ exp ξ /a ).exp a γc ) / ξ + c ) ξ c /a ) / c aγc ) /γ/ ξ exp ξ /a ) =.exp a γc ) / ξ + c c /a ) / )exp ξ c ac ) /ξ /a ) { ) } =.exp a γc ) / + / c a ξ exp ξ /a ). c In our particular problem which corresponds to Theorem of Bitouzé, Laurent and Massart )), a = from Bitouzé, Laurent and Massart ), proof of Theorem, pages -), and γ = K is given by Theorem above, so we calculate a γc ) / + / c a c.. Note that the bound of ) holds trivially for λ = ξ. since the right side with the above choice of C is for all such values of λ; thus the exponential bound holds trivially for λ = ξ < bc.. This completes the proof of Proposition.. Proof of Theorem Proof. Let G denote the collection of all distribution functions concentrated on 0,] i.e. corresponding to the distributions of random variables V with V 0,] with probability ). As shown in van der Vaart and Wellner ), we can reduce to finding a bound for log N ], G,L r λ)) for the uniform probability measure λ on 0,] i.e. Lebesgue measure). We say F L is a left bracket for F G if F L x) Fx) for all x 0,], and F L F λ,r ; similarly, F R is a right bracket for F G if F R x) Fx) for all x 0,], and F R F λ,r. To obtain an upper bound for the bracketing number for the class G, the key is to define left and right brackets and count the number require to cover G. As in the proof

12 0 0 0 Wellner/Exponential Bound for Kaplan - Meier of Theorem.. in van der Vaart and Wellner ) which is based on van de Geer ) who, in turn, used results and methods of Birman and Solomak )), the idea here is based on different levels of partitioning of the unit interval 0,]. At each level of partitioning, the class G is partitioned into finitely many subsets. Then we construct left and right brackets, so that they have size proportional to), and find an upper bound for the total number of brackets. Fix > 0. Let c = /r. Fix F G. Let P 0 be the trivial) partition of the unit interval 0 = u 0) 0 < u 0) =. The i-th partition P i of 0,] is of the form 0 = u i) 0 < u i) < < u i) =. n i) Given a partition P i, define i i F) = ] max Fu i) n i) ) Fui) ) u i) u i) )/r). To form the next partition P i+ P i, find all the intervals in u i),ui) ] in the partition P i that satisfy ] Fu i) ) Fui) ) u i) u i) )/r c i. ) Then, the partition P i+ is obtained from P i by dividing all the intervals u i),ui) ) that satisfy ) into two halves of equal length. The partition P i+ is obtained by forming S i+) = u i+),ui+) ], where {,,...,n i+) }. Let u i+) ] be an interval in the partition P i+ that is contained in,ui+) the interval u i),ui) ] in partition P i. If ) is not satisfied for an interval u i),ui) ], then Fu i+) = If ) is satisfied for u i) and It follows that ] ) Fu i+) ) u i+) u i+) )/r ] Fu i) ) Fui) ) u i) u i) )/r i+ c i.,ui) ) in P i, then u i+) Fu i+) ) Fu i+) ) Fui) ) Fui) ). Fu i+) c This shows again that i+ c i. ] ) Fu i+) ) u i+) u i+) )/r ] Fu i) ) Fui) ) u i) u i) )/r. u i+) ) = ui) u i),

13 0 0 0 Wellner/Exponential Bound for Kaplan - Meier On the other hand, let u i),ui) ] be an interval in the partition P i that satisfies ) and ] i = Fu i) ) Fui) ) u i) u i) )/r. It follows from the definition of i+ that i+ Hence, we have F u i) + F u i) + u i) ) ) ] F u i) u i) ) F u i) + u i) )] u i) ui) ] = c Fu i) ) Fui) ) u i) u ) i) /r = ci. + u i) ) /r u i) + u i) )/r i+ c i i+. ) Let n i n i F) = n i) be the number of intervals in the partition P i. Let s i = n i+ n i be the number of intervals in P i that are divided to obtain P i+. It follows from the definitions of s i, i and Hölder s inequality with p = r + )/r and q = r + ) that s i c i ) r Therefore Consequently, n i) = = ] Fu i) r ) Fui) ) u i) u i) ) n i) ) Fu i) ) F ku i) ) =. i n = i + = since i = Note that, by ), i s i i + = s i c i ) r. i c i ) r = c i ) r ic r c i i + i + c r r n i) = u i) u i) i = 0 = i /) i. i + c i c. ) c i ) r

14 0 0 0 This implies that Thus, i n = Then, we have T i ) It follows that Hence, i = Wellner/Exponential Bound for Kaplan - Meier c i ) r c )r c )r r Ti = + = T i i = i = r i. ) r i T i r i. ) ) ) i ) i i ) i ) i = i ) i ) ] C r). ) n C r) r i. ) The above partitions P 0 P are generated by a fixed function F G. Two monotone functions F and G are said to be equivalent at level i if their partitions up to the ith level are the same. As we continue the partitioning process, the class G is equivalently partitioned into a finite number of subsets. Let L denote the first such level of partition that L F) r for every F in that subset of G. For each subset and i, define i = sup i F), F where F ranges over the subset of G in the final level partition. Note that the number of intervals n 0 n n i only depends on the sequence of partitions P 0 P P i. Thus, the i in the inequality ) can be replaced by i. To obtain the upper bound for the bracketing number N ], G,L r λ)), we first find an upper bound for the total number of different final partitions. By definition of the L th level partition, we know that there exists a function G G such that L G) r > at the L )th level partition. This implies that L G) r /. Then it follows from ) that L = n C r) r L C r)..

15 0 0 0 Since n L n L, then we have Wellner/Exponential Bound for Kaplan - Meier = n 0 n n L C, ) where C Cr) = C r) = /) /)]. To count the number of sequences n 0 n n L C/, note that this is equivalent to picking L numbers out of {,,,..., C/ }, and this is easily seen to be is C/ ) L, which is bounded by C/. For a given sequence n 0 n n L, the number of ways to obtain P i+ from P i is n i ) s i which is bounded by n i. Thus, the total number of different final partitions of the form P 0 P P L when F ranges over G is bounded by C/ n n L C/ since L i= n i C/. We now define the bracket F l,f r ] for a fixed function F G in the final stage partition P L. For u 0,], let and F l u) = n L) = F r u) = n L) = { ) F u L) / { F u L) ) / S L) S L) } u), } u). For u S L), it follows from ), the definitions of the brackets, and L that ) F r u) F l u) F u L) ) + F u L) ) ) = + F u L) F u L) L + ) /r + u L) u L) L u L) u L) ) /r.

16 0 0 0 Wellner/Exponential Bound for Kaplan - Meier Thus, we have, using the inequality x + y r r { x r + y r }, F r u) F l u) r λ,r n L) + L = u L) u L) )/r { n L) r ) r + L ) r = n L) = r )r = u L) ] λs L) ) + L ) r r u L) λs L) ) n L) = ]} λs L) ) λs L) ) u L) u L) ] { = r ) r + L ) r n L)} { ]} C r ) r + = r r + C) r ) by ). Here we have also used the fact that λ is the uniform distribution, and hence u L) u L) = λsl) ). The inequality ) implies that the brackets F l,f r ] have the right size. Now we count how many brackets can be constructed on the final stage partition P L when F ranges over the subset of G for a given sequence of partitions P 0 P P L. Let Fu ) l =, r = Fu ) As argued in van der Vaart and Wellner ), page, the number of left brackets for a given final partition is at most L C/ with L = /c + ) = /r +), and similarly for the number of right brackets. Thus we conclude that N r ] r + C) ] ) /r, G,Lr λ) or, equivalently That is,. C/ L C/ L C/ = L ) C/, log N r ] r + C) ] ) /r, G,Lr λ) log N ], G,L r λ)) K, C logl ). where K = Kr) C logl )] r r + C) ] /r. Combining this with the conclusion of the reduction step at the beginning of the proof yields the stated bound.

17 Acknowledgements Wellner/Exponential Bound for Kaplan - Meier I owe thanks to John Crowley for stimulating my interest in the Kaplan- Meier estimator during conversations in Seattle in - concerning his work with Norman Breslow on the asymptotic theory of the Kaplan-Meier estimator. I also owe thanks to Richard Gill for his many probing questions and challenges about censored data over the years, and to Shuguang Song for help with the computation of the constants in Theorem. References Birman, M. Š. and Solomak, M. Z. ). Piecewise polynomial approximations of functions of classes W α p. Mat. Sb. N.S.) ). Bitouzé, D., Laurent, B. and Massart, P. ). A Dvoretzky-Kiefer- Wolfowitz type inequality for the Kaplan-Meier estimator. Ann. Inst. H. Poincaré Probab. Statist.. Breslow, N. and Crowley, J. ). A large sample study of the life table and product limit estimates under random censorship. Ann. Statist.. Burke, M. D., Csörgő, S. and Horváth, L. ). Strong approximations of some biometric estimates under random censorship. Z. Wahrsch. Verw. Gebiete. Chen, K. and Ying, Z. ). A counterexample to a conecture concerning the Hall-Wellner band. Ann. Statist.. Csörgő, S. ). Universal Gaussian approximations under random censorship. Ann. Statist.. Csörgő, S. and Horváth, L. ). The rate of strong uniform consistency for the product-limit estimator. Z. Wahrsch. Verw. Gebiete. Csörgő, S. and Horváth, L. ). Confidence bands from censored samples. Canad. J. Statist.. Dvoretzky, A., Kiefer, J. and Wolfowitz, J. ). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist.. Gill, R. ). Large sample behaviour of the product-limit estimator on the whole line. Ann. Statist.. Gill, R. D. 0). Censoring and stochastic integrals, vol. of Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam.

18 0 0 0 Wellner/Exponential Bound for Kaplan - Meier Gill, R. D. ). Glivenko-Cantelli for Kaplan-Meier. Math. Methods Statist.. Gillespie, M. J. and Fisher, L. ). Confidence bands for the Kaplan- Meier survival curve estimate. Ann. Statist. 0. Giné, E. and Guillou, A. ). Laws of the iterated logarithm for censored data. Ann. Probab Gu, M. ). The Chung-Smirnov law for the product-limit estimator under random censorship. Chinese Ann. Math. Ser. B. A Chinese summary appears in Chinese Ann. Math. Ser. A ), no.,. Gu, M. G. and Lai, T. L. 0). Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation. Ann. Probab.. Hall, W. J. and Wellner, J. A. 0). Confidence bands for a survival curve from censored data. Biometrika. Hollander, M., McKeague, I. W. and Yang, J. ). Likelihood ratio-based confidence bands for survival functions. J. Amer. Statist. Assoc.. Hollander, M. and Peña, E. ). Families of confidence bands for the survival function under the general random censorship model and the Koziol-Green model. Canad. J. Statist.. Kaplan, E. L. and Meier, P. ). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc.. Massart, P. 0). The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab.. McKeague, I. W. and Zhao, Y. 00). Width-scaled confidence bands for survival functions. Statist. Probab. Lett.. Meier, P. ). Estimation of a distribution function from incomplete observations. In Perspectives in probability and statistics papers in honour of M. S. Bartlett on the occasion of his th birthday). Applied Probability Trust, Univ. Sheffield, Sheffield,. Nair, V. N. ). Plots and tests for goodness of fit with randomly censored data. Biometrika. Pollard, D. 0). Empirical processes: theory and applications. NSF- CBMS Regional Conference Series in Probability and Statistics,, Institute of Mathematical Statistics, Hayward, CA.

19 0 0 0 Wellner/Exponential Bound for Kaplan - Meier Shorack, G. R. and Wellner, J. A. ). Empirical processes with applications to statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York. Stute, W. a). Convergence of the Kaplan-Meier estimator in weighted sup-norms. Statist. Probab. Lett. 0. Stute, W. b). Strong and weak representations of cumulative hazard function and Kaplan-Meier estimators on increasing sets. J. Statist. Plann. Inference. Stute, W. and Wang, J.-L. ). The strong law under random censorship. Ann. Statist.. van de Geer, S. ). The entropy bound for monotone functions. Tech. Rep. Report -, University of Leiden. van de Geer, S. ). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist.. van de Geer, S. A. 000). Applications of empirical process theory, vol. of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. van der Vaart, A. W. and Wellner, J. A. ). Weak convergence and empirical processes. Springer Series in Statistics, Springer-Verlag, New York. With applications to statistics. Wang, J. G. ). A note on the uniform consistency of the Kaplan-Meier estimator. Ann. Statist.. Ying, Z. ). A note on the asymptotic properties of the product-limit estimator on the whole line. Statist. Probab. Lett.. Jon A. Wellner University of Washington Department of Statistics Box Seattle, WA - aw@stat.washington.edu