The link between the nonparametric estimator of the crude cumulative incidence of a competing risk and the Kaplan-Meier estimator is exploited. The equivalence of the nonparametric crude cumulative incidence to an inverse-probability-of-censoring weighted average of the sub-distribution function is proved. The link between the estimation of crude cumulative incidence curves and Gray's family of nonparametric tests is considered. The crude cumulative incidence is proved to be a Kaplan-Meier like estimator based on the sub-distribution hazard, i.e. the quantity on which Gray's family of tests is based. A standard probabilistic formalism is adopted to have a note accessible to applied statisticians.
Antolini, Laura; Biganzoli, Elia Mario; and Boracchi, Patrizia, "Crude Cumulative Incidence in the form of a Horvitz-Thompson like and Kaplan-Meier like Estimator" (October 2006). COBRA Preprint Series. Working Paper 10.