*J. Japan Statist. Soc.,* Vol. 32 (No. 1), pp. 77-93, 2002

## Asymptotic Properties of Aalen-Johansen Integrals for Competing Risks Data

Akio Suzukawa

**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*_{n}^{(j)} 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*_{n}^{(j)} = *j* *dF*_{n}^{(j)} are investigated. Let *F* be the overall lifetime distribution. We show that for any *F*-integrable function *j*, the Aalen-Johansen integral *s*_{n}^{(j)} converges almost surely as *n* . It is also shown that under some mild integrability assumptions for *j*, the joint distribution of *s*_{n}^{(j)}'s for all causes is asymptotically multivariate normal.

*Key words and phrases*:
Aalen-Johansen estimator, cumulative incidence function, Kaplan-Meier integral.

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