ASYMPTOTICALLY LOCAL MINIMAX ESTIMATION OF INFINITELY SMOOTH DENSITY WITH CENSORED DATA

EDUARD BELITSER^{1}, AND BORIS LEVIT^{2}

1Department of Mathematics, Vrije Universiteit, de Boelelaan 1081a, 1081 HV Amsterdam, the Netherlands

2Mathematical Institute, Utrecht University, Budapestlaan 6, 3584 CD Utrecht, the Netherlands

(Received February 9, 1998; revised September 28, 1999)

Abstract. The problem of the nonparametric minimax estimation of an infinitely smooth density at a given point, under random censorship, is considered. We establish the exact asymptotics of the local minimax risk and propose the efficient kernel-type estimator based on the well known Kaplan-Meier estimator.

Key words and phrases: Efficient estimator, local minimax risk, Kaplan-Meier estimator, kernel, random censorship.