Laws of the Iterated Logarithm for Censored Data

Abstract

First- and second-order laws of the iterated logarithm are obtained for both the Nelson–Aalen and the Kaplan–Meier estimators in the random censorship model, uniform up to a large order statistic of the censored data. The rates for the first-order processes are exact except for constants. The LIL for the second-order processes (where one subtracts a linear, empirical process, term from the difference between the original process and the estimator), uniform over fixed intervals, is also proved. Somewhat surprisingly, there is a certain degree of proof unification for fixed and variable intervals in the second-order results for the Nelson–Aalen estimator. No assumptions are made on the distribution of the censoring variables and only continuity of the distribution function of the original variables is assumed for the results on the Kaplan–Meier estimator.