Nonparametric Least squares estimators for interval censoring

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].