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 sare 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 √ nsn 's for all causes is asymptotically multivariate normal.
N. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B 46, 440-464) introduced a class of multinomial goodness-of-fit statistics R a based on power divergence. All R a have the same chi-square limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of R a under local alternatives. We obtain an expression of approximation for the distribution of R a under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of R a . We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of R a . We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance.
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