We study the uniform convergence rate of the nonparametric maximum likelihood estimator (MLE) for the sub-distribution functions in the current status data with competing risks model. It is known that the MLE have $L^2$-norm convergence rate $O_P(n^{-1/3})$ in the absolutely continuous case, but there is no arguments for the same rate of uniform convergence. We specify conditions for the uniform convergence rate $O_P(n^{-1/3}\log^{1/3} n)$ of the MLE for the sub-distribution functions of competing risks on finite intervals. The obtained result refines known uniform convergence rate in the particular case of current status data. The main result is applied in order to get the uniform convergence rate of the MLE for the survival function of failure time in the current status right-censored data model.

中文翻译：

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具有竞争风险的当前状态数据的非参数最大似然估计器的统一收敛率

我们研究了具有竞争风险模型的当前状态数据中子分布函数的非参数最大似然估计器（MLE）的统一收敛率。已知 MLE 在绝对连续的情况下具有 $L^2$-范数收敛率 $O_P(n^{-1/3})$，但是对于相同的均匀收敛率没有争论。我们为有限区间上竞争风险的子分布函数指定了 MLE 的统一收敛率 $O_P(n^{-1/3}\log^{1/3} n)$ 的条件。得到的结果在当前状态数据的特定情况下改进了已知的统一收敛率。应用主要结果是为了获得当前状态右删失数据模型中失效时间生存函数的 MLE 的统一收敛率。