Nonparametric Estimation for a Log-concave Distribution Function with Interval-censored Data

TL;DR

This paper develops a new nonparametric maximum likelihood estimator for log-concave distribution functions based on interval-censored data, providing theoretical guarantees and an efficient computational algorithm.

Contribution

It introduces a generalized framework for estimating log-concave distribution functions with interval-censored data, relaxing previous assumptions and offering a practical implementation.

Findings

The estimator is proven to exist, be unique, and be consistent.

Numerical studies show improved performance over unconstrained estimators.

The method balances efficiency and robustness under the log-concavity assumption.

Abstract

We consider the nonparametric maximum likelihood estimation for the underlying event time based on mixed-case interval-censored data, under a log-concavity assumption on its distribution function. This generalized framework relaxes the assumptions of a log-concave density function or a concave distribution function considered in the literature. A log-concave distribution function is fulfilled by many common parametric families in survival analysis and also allows for multi-modal and heavy-tailed distributions. We establish the existence, uniqueness and consistency of the log-concave nonparametric maximum likelihood estimator. A computationally efficient procedure that combines an active set algorithm with the iterative convex minorant algorithm is proposed. Numerical studies demonstrate the advantages of incorporating additional shape constraint compared to the unconstrained nonparametric maximum likelihood estimator. The results also show that our method achieves a balance between efficiency and robustness compared to assuming log-concavity in the density. An R package iclogcondist is developed to implement our proposed method.