The limit distribution of the nonparametric maximum likelihood estimator for interval censored data with more than one observation time per unobservable observation, is still unknown in general. For the so-called separated case, where one has observation times which are at a distance larger than a fixed $\epsilon>0$, the limit distribution was derived in [4]. For the non-separated case there is a conjectured limit distribution, given in [9], Section 5.2 of Part 2. But the findings of the present paper suggest that this conjecture may not hold. We prove consistency of a closely related nonparametric isotonic least squares estimator and give a sketch of the proof for a result on its limit distribution. We also provide simulation results to show how the nonparametric MLE and least squares estimator behave in comparison. Moreover, we discuss a simpler least squares estimator that can be computed in one step, but is inferior to the other least squares estimator, since it does not use all information. For the simplest model of interval censoring, the current status model, the nonparametric maximum likelihood and least squares estimators are the same. This equivalence breaks down if there are more observation times per unobservable observation. The computations for the simulation of the more complicated interval censoring model were performed by using the iterative convex minorant algorithm. They are provided in the GitHub repository [6].

翻译：对于每个不可观测观测值具有多个观测时间的区间删失数据，其非参数最大似然估计量的极限分布通常仍属未知。在所谓的分离情形（即观测时间间隔大于固定值ε>0）下，文献[4]已推导出极限分布。对于非分离情形，文献[9]第二部分第5.2节提出了一个推测的极限分布，但本文的研究结果表明该推测可能不成立。我们证明了一种密切相关的非参数保序最小二乘估计量的一致性，并给出了其极限分布结果的证明概要。通过仿真实验，我们对比展示了非参数最大似然估计量与最小二乘估计量的实际表现。此外，我们讨论了一种可单步计算的简化最小二乘估计量，但由于其未能利用全部信息，性能逊于前述最小二乘估计量。在区间删失的最简模型（当前状态模型）中，非参数最大似然估计量与最小二乘估计量具有等价性，但当每个不可观测观测值对应更多观测时间时，这种等价性将不再成立。复杂区间删失模型的仿真计算通过迭代凸小化算法实现，相关代码已发布于GitHub仓库[6]。