Study of the Nonparametric Kaplan-Meier Estimator of the Cumulative Incidence Function in Competiting Risks

Abstract

In this paper, inspired by the estimator of the cumulated specific incidence proposed by Marubini and Vasecchi [23], we obtain the Kaplan-Meier estimator of the survival function of all causes of death combined by summing the estimator of fj(t) (j ∈ {1, . . . , m}) obtained by plug-in. We establish that the Kaplan-Meier estimator of the survival function overestimates the cumulative incidence in the presence of competitive events. By making the product of all the contributions of the studied system, we establish the likelihood function of the specific risk function for the competitive risk model. Finally, under the assumptions of Dinse and Larson [13], and using the delta-method, we establish the variance of the cumulative incidence function in competiting risks.

Asymptotic Normality of the Nelson-Aalen and the Kaplan-Meier Estimators in Competing Risks

“…The model of competing risks has been widely studied in the literature, see e.g., Heckman and Honoré [1], Commenges [2], Com-nougué [3], Fine and Gray [4], Crowder [5], Fermanian [6], Latouche, A. [7], Geffray [8], Belot [9], Njamen and Ngatchou ( [10], [11]), Njamen ([12], [13]). In most approaches, the competing risks are assumed to be either all independent or all dependent.…”

“…In Sect. 3.2, we give limit law of Kaplan-Meier's nonparametric estimator in competing risks as defined in Njamen and Ngatchou [10] and Njamen [13]. In Section 4, we give the trust Bands, including the Hall-Wellner trust Bands and the Nair precision equal bands.…”