# THE EMPIRICAL LIL FOR THE KAPLAN-MEIER INTEGRAL PROCESS

• Bae, Jong-Sig (Department of Mathematics, SungKyunKwan University) ;
• Kim, Sung-Yeun (Department of Mathematics, SungKyunKwan University)
• 발행 : 2003.05.01

#### 초록

We prove an empirical LIL for the Kaplan-Meier integral process constructed from the random censorship model under bracketing entropy and mild assumptions due to censoring effects. The main method in deriving the empirical LIL is to use a weak convergence result of the sequential Kaplan-Meier integral process whose proofs appear in Bae and Kim [2]. Using the result of weak convergence, we translate the problem of the Kaplan Meier integral process into that of a Gaussian process. Finally we derive the result using an empirical LIL for the Gaussian process of Pisier [6] via a method adapted from Ossiander [5]. The result of this paper extends the empirical LIL for IID random variables to that of a random censorship model.

#### 참고문헌

1. J. Korean Math. Soc. v.33 no.4 An empirical LIL for stationary martingale differences : An invariance principle approach J.Bae
2. Bull. Austral. Math. Soc. The uniform CLT for the Kaplan-Meier integral process under bracketing entropy J.Bae;S.Kim
3. Z. Wahrsch. verw. Gebiete v.62 Invariance principles for sums of Banach space valued random elements and empirical processes R.M.Dudley;W.Philipp https://doi.org/10.1007/BF00534202
4. Ann. Probab v.8 Log log law for empirical measures J.Kuelbs;R.M.Dudley https://doi.org/10.1214/aop/1176994716
5. Ann. Probab. v.15 A central limit theorem under metric entropy with L₂bracketing M.Ossiander https://doi.org/10.1214/aop/1176992072
6. Seminaire Maurey-Schwarz 1975-1976 exposes Nos. 3 et 4 Le theoreme de la limite centrale et la loi du logarithme itere dans les espaces de Banach G.Pisier
7. Springer series in Statistics Convergence of stochastic processes D.Pollard
8. Ann. Statist. v.23 The central limit theorem under random censorship W.Stute https://doi.org/10.1214/aos/1176324528
9. Springer series in Statistics Weak convergence and empirical processes with applications to statistics A.W.Van der Vaart;J.A.Wellner