THE 2ND-ORDER OPTIMALITY OF A SMOOTHED KAPLAN-MEIER ESTIMATOR

In this paper we first consider a real-valued non-linear differentiabl e functional T defined on a set of probability measures P. The set P i s called a statistical model for i.i.d. observations. Extending an app roach of B. Ya. Levit we define the notion of integrability in a certa in direction for the statistical model and prove a minimax risk theore m for estimators of T. The main part of the paper uses the theorem to discuss the second order optimality of the Kaplan-Meier estimator and, in the case that the underlying distribution function is supposed to be smooth, of a smoothed version of the Kaplan-Meier estimator.