Nonparametric Least Squares Estimators for Interval Censoring

TL;DR

This paper investigates the limit distribution of nonparametric estimators for interval censored data, proving consistency of a least squares estimator, comparing it with the MLE, and discussing computational methods and limitations in observing the conjectured limit behavior.

Contribution

It introduces a consistent nonparametric isotonic least squares estimator for interval censoring and compares its performance with the MLE, including computational aspects and theoretical insights.

Findings

Limit distribution remains unclear for large samples.

The least squares estimator is consistent and comparable to the MLE.

Computational methods using iterative convex minorant are provided.

Abstract

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 positive epsilon, the limit distribution was derived in [5]. For the non-separated case there is a conjectured limit distribution, given in [10], Section 5.2 of Part 2. Whether this conjecture holds is still unknown, but the present paper shows that for sample sizes 1000 and 10,000 this limit behavior is still not clearly seen. We prove consistency of a related nonparametric isotonic least squares estimator and sketch of the proof for 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 [7].