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.
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