A new function for the competing risks model, the conditional cumulative hazard function, is introduced, from which the conditional distribution of failure times of individuals failing due to cause j can be studied. The standard Nelson-Aalen estimator is not appropriate in this setting, as population membership (mark) information may be missing for some individuals owing to random right-censoring. We propose the use of imputed population marks for the censored individuals through fractional risk sets. Some asymptotic properties, including uniform strong consistency, are established. We study the practical performance of this estimator through simulation studies and apply it to a real data set for illustration.

中文翻译：

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缺失人口标记的条件累积危害的非参数估计

引入了竞争风险模型的一个新函数，条件累积风险函数，从中可以研究由于原因 j 导致失败的个体失败次数的条件分布。标准的 Nelson-Aalen 估计量不适用于此设置，因为由于随机右删失，某些个体可能会丢失总体成员资格（标记）信息。我们建议通过分数风险集对审查个人使用估算的人口标记。建立了一些渐近性质，包括均匀强一致性。我们通过模拟研究来研究该估计器的实际性能，并将其应用于真实数据集以进行说明。