Uniform convergence rate of nonparametric maximum likelihood estimator for the current status data with competing risks

TL;DR

This paper establishes the uniform convergence rate of the nonparametric maximum likelihood estimator for sub-distribution functions in current status data with competing risks, refining previous results and applying to survival function estimation.

Contribution

It provides the first uniform convergence rate of the MLE in the competing risks current status model, including conditions and applications.

Findings

Uniform convergence rate is $O_P(n^{-1/3}\log^{1/3} n)$ on finite intervals.

Refines known $L^2$-norm convergence results to uniform convergence.

Applied to derive convergence rate for survival function in right-censored data.

Abstract

We study the uniform convergence rate of the nonparametric maximum likelihood estimator (MLE) for the sub-distribution functions in the current status data with competing risks model. It is known that the MLE have $L^2$-norm convergence rate $O_P(n^{-1/3})$ in the absolutely continuous case, but there is no arguments for the same rate of uniform convergence. We specify conditions for the uniform convergence rate $O_P(n^{-1/3}\log^{1/3} n)$ of the MLE for the sub-distribution functions of competing risks on finite intervals. The obtained result refines known uniform convergence rate in the particular case of current status data. The main result is applied in order to get the uniform convergence rate of the MLE for the survival function of failure time in the current status right-censored data model.