ESTIMATION OF A MONOTONE DENSITY OR MONOTONE HAZARD UNDER RANDOM CENSORING

Consider non-parametric estimation of a decreasing density function f under the random (right) censorship model. Alternatively, consider est imation of a monotone increasing (or decreasing) hazard rate lambda ba sed on randomly right censored data, We show that the non-parametric m aximum likelihood estimator of the density f(introduced by Laslett, 19 82) is asymptotically equivalent to the estimator obtained by differen tiating the least concave majorant of the Kaplan-Meier estimator, the non-parametric maximum likelihood estimator of the distribution functi on Fin the larger model without any monotonicity assumption, A similar result is shown to hold for the nan-parametric maximum likelihood est imator of an increasing hazard rate lambda: the non-parametric maximum likelihood estimator of lambda (introduced in the uncensored case by Prakasa Rao, 1970) is asymptotically equivalent to the estimator obtai ned by differentiation of the greatest convex minorant of the NeIson-A alen estimator, the non-parametric maximum likelihood estimator of the cumulative hazard function A in the larger model without any monotoni city assumption, in proving these asymptotic equivalences, we also est ablish the asymptotic distributions of the different estimators at a f ixed point at which the monotonicity assumption is strictly satisfied,