## Estimation of a monotone density or monotone hazard under random censoring.(English)Zbl 0827.62032

Summary: Consider nonparametric estimation of a decreasing density function $$f$$ under the random (right) censoring model. Alternatively, consider estimation of a monotone increasing (or decreasing) hazard rate $$\lambda$$ based on randomly right censored data. We show that the nonparametric maximum likelihood estimator of the density $$f$$ is asymptotically equivalent to the estimator obtained by differentiating the least concave majorant of the Kaplan-Meier estimator, the nonparametric maximum likelihood estimator of the distribution function $$F$$ in the larger model without any monotonicity assumption.
A similar result is shown to hold for the nonparametric maximum likelihood estimator of an increasing hazard rate $$\lambda$$: the nonparametric maximum likelihood estimator of $$\lambda$$ is asymptotically equivalent to the estimator obtained by differentiation of the greatest convex minorant of the Nelson-Aalen estimator, the nonparametric maximum likelihood estimator of the cumulative hazard function $$\Lambda$$ in the larger model without any monotonicity assumption.
In proving these asymptotic equivalences, we also establish the asymptotic distributions of the different estimators at a fixed point at which the monotonicity assumption is strictly satisfied.

### MSC:

 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62M09 Non-Markovian processes: estimation 62M99 Inference from stochastic processes