An improved nonparametric estimator of sub-distribution function for bivariate competing risk models

For competing risks data, it is of interest to estimate the sub-distribution function of a particular failure event, which is the failure probability in the presence of competing risks. However, if multiple failure events per subject are available, estimation procedures become challenging even for the bivariate case. In this paper, we consider nonparametric estimation of a bivariate sub-distribution function, which has been discussed in the related literature. Adopting a decision-theoretic approach, we propose a new nonparametric estimator which improves upon an existing estimator. We show theoretically and numerically that the proposed estimator has smaller mean square error than the existing one. The consistency of the proposed estimator is also established. The usefulness of the estimator is illustrated by the salamander data and mouse data.