Multiplicative censoring, renewal processes, deconvolution and decreasing density: Nonparametric estimation

Abstract

This paper derives the nonparametric maximum likelihood estimate for a lifetime distribution G , under the following ‘multiplicative-censorship’ model: X 1 ,…, X m are complete uncensored observations from G , and Y 1 ,…, Y n are incomplete observations from G . The incompleteness of the Y i 's is assumed to come from the following censoring mechanism. For each Y i , there exists an unobserved ‘parent-observation’ Z i distributed according to G , and Y i is the product of Z i and an independent uniform (0, 1) random variable; i.e. conditional on Z i , Y i is distributed uniformly over (0, Z i ). We show that this model generalizes several well studied statistical problems such as estimating a distribution function under a decreasing density constraint, nonparametric deconvolution of an exponential random variable, and an estimation problem in renewal processes. We also point out the potential usefulness of this model as a framework for informative censoring.