Title: Nonparametric Estimation in Uniform Deconvolution and Interval Censoring Show Chinese title

Title: 非参数估计在一致去卷积和区间审查中的应用

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. Show Chinese abstract

Abstract: 在一致去卷积问题中，人们感兴趣的是估计一个非负随机变量的分布函数 $F_0$，基于带有均匀噪声的样本。 在这种设定下，非参数最大似然估计量的一个奇特且尚未被充分理解的现象是 $F_0(1)=1$ 和 $F_0(1)<1$ 这两种情况之间的二分现象。 如果 $F_0(1)=1$，最大似然估计（MLE）可以通过一种直接的方式计算，并且可以利用与所谓的当前状态问题的联系推导出其渐近点态行为。 然而，如果 $F_0(1)<1$，则需要迭代程序来计算它，并且非参数最大似然估计量的渐近点态行为尚不清楚。 在本文中，我们描述了这个问题，将其与区间删失问题以及Groeneboom（2024年）研究的更一般模型联系起来，针对 $F_0(1)<1$ 的情形提出了两个竞争性的自然出现的猜想。 基于渐近论据和平滑函数理论以及广泛的模拟，我们倾向于其中一个猜想。