Density Estimation for Lifetime Distributions Under Semi-parametric Random Censorship Models

Abstract

We derive product limit estimators of survival times and failure rates for randomly right censored data as the numerical solution of identifying Volterra integral equations by employing explicit and implicit Euler schemes. While the first approach results in some known estimators, the latter leads to a new general type of product limit estimator. Plugging in established methods to approximate the conditional probability of the censoring indicator given the observation, we introduce new semi-parametric and presmoothed Kaplan-Meier type estimators. In the case of the semi-parametric random censorship model, i.e. the latter probability belonging to some parametric family, we study the strong consistency and asymptotic normality of some linear functionals based on the proposed estimator. Assuming that the underlying random variable admits a probability density, we define semi-parametric and presmoothed kernel estimators of the density and the hazard rate for randomly right censored data, which rely on the newly derived estimators of the survival function. We determine exact rates of pointwise and uniform convergence as well as the limiting distribution.