SCIPRESS J. Japan Statist. Soc.

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 Fn(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 sn(j) = j dFn(j) are investigated. Let F be the overall lifetime distribution. We show that for any F-integrable function j, the Aalen-Johansen integral sn(j) converges almost surely as n . It is also shown that under some mild integrability assumptions for j, the joint distribution of sn(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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