ASYMPTOTIC PROPERTIES OF AALEN-JOHANSEN INTEGRALS FOR COMPETING RISKS DATA

Abstract

This paper considers the competing risks problem with randomly right-censored data. Let F^(j)(t) be the cause-specific cumulative incidence function of a cause j, which is the probability of death due to a cause j by time t in the presence of other acting causes. The Aalen-Johansen estimator F^(j)_n is a nonparametric maximum likelihood estimator of F^(j). Under the assumption that all F^(j)’s and a censoring distribution are continuous, asymptotic properties of the Aalen-Johansen integral s^(j)_n=∫φdF^(j)_n are investigated. Let F be the overall lifetime distribution. We show that for any F-integrable function φ, the Aalen-Johansen integral s^(j)_n converges almost surely as n→∞. It is also shown that under some mild integrability assumptions for φ, the joint distribution of √¯ns^(j)_n’s for all causes is asymptotically multivariate normal.