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

Huang, Jian; Wellner, Jon A.

Scand. J. Stat. 22, No. 1, 3-33 (1995).

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

Keywords:

argmax continuous mapping; empirical process; hazard function; weak approximation; decreasing density function; randomly right censored data; nonparametric maximum likelihood estimator; least concave majorant; Kaplan-Meier estimator; monotonicity; increasing hazard rate; greatest convex minorant; Nelson-Aalen estimator; asymptotic equivalences