Multiplicative Censoring, Renewal Processes, Deconvolution and Decreasing Density: Nonparametric Estimation

This paper derives the nonparametric maximum likelihood estimate for a lifetime distribution G, under the following ‘multiplicative-censorship’ model: X1,…, Xm are complete uncensored observations from G, and Y1,…, Yn are incomplete observations from G. The incompleteness of the Yi's is assumed to come from the following censoring mechanism. For each Yi, there exists an unobserved ‘parent-observation’ Zi distributed according to G, and Yi is the product of Zi and an independent uniform (0, 1) random variable; i.e. conditional on Zi, Yi is distributed uniformly over (0, Zi). 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.