Nonparametric Estimation in Uniform Deconvolution and Interval Censoring

Abstract

In the uniform deconvolution problem one is interested in estimating the distribution function $F_0$ of a nonnegative random variable, based on a sample with additive uniform noise. A peculiar and not well understood phenomenon of the nonparametric maximum likelihood estimator in this setting is the dichotomy between the situations where $F_0(1)=1$ and $F_0(1)<1$. If $F_0(1)=1$, the MLE can be computed in a straightforward way and its asymptotic pointwise behavior can be derived using the connection to the so-called current status problem. However, if $F_0(1)<1$, one needs an iterative procedure to compute it and the asymptotic pointwise behavior of the nonparametric maximum likelihood estimator is not known. In this paper we describe the problem, connect it to interval censoring problems and a more general model studied in Groeneboom (2024) to state two competing naturally occurring conjectures for the case $F_0(1)<1$. Asymptotic arguments related to smooth functional theory and extensive simulations lead us to to bet on one of these two conjectures.