Non-parametric estimation under progressive censoring

Abstract

We consider non-parametric estimation of cumulative hazard functions and reliability functions of progressively type-II right censored data. As shown in the book of Balakrishnan and Aggarwala (Progressive Censoring, Birkhäuser, Basel, 2000), many results of classical order statistics can be generalized to this kind of statistics. These authors proposed also many inferential methods for parametric models. In this paper we show that non-parametric maximum likelihood estimators (NPMLE) may also be derived under such censoring schemes. These estimators are obtained in a reliability context but they can also be extended to arbitrary continuous distribution functions. Since the large sample properties of the NPMLE depend on counting processes based upon generalized order statistics that are generated by progressive censoring, we need to establish some basic properties of these processes (e.g. martingales properties and weak consistency). Finally, the non-parametric estimator of the reliability is compared with two parametric estimators for a real data set and additionally, some Monte-Carlo simulations are provided.