The cumulative incidence function (CIF) plays an important role in the comparison of competing risks in a competing risks model. Its value at time $t$ is the probability of failure by time $t$ from a particular type of risk in the presence of other risks. In this paper we consider the estimation of two CIFs, $F_1$ and $F_2$, corresponding to two competing risks when the ratio $R\left(t\right)\equiv F_1\left(t\right)∕F_2\left(t\right)$ is nondecreasing in $t>0$. First, we derive their nonparametric maximum likelihood estimators (NPMLE) of these CIFs in the continuous case under this order constraint and show that they are inconsistent. We then develop projection-type estimators that are uniformly strongly consistent and study the weak convergence of the resulting processes. Through simulations, we compare the finite sample performance of the NPMLEs and our estimators and show that our estimators outperform them in general in terms of mean square error at all the scenarios that we consider. We also develop a test for the presence of this order constraint and extend all these results to the censored case. To illustrate the applicability of the theory we develop, we provide a real life example.

累积发生率函数 (CIF) 在竞争风险模型中的竞争风险比较中起着重要作用。其当时的价值$吨$ 是按时间失败的概率 $吨$从存在其他风险的特定类型的风险。在本文中，我们考虑两个 CIF 的估计，$F_1$ 和 $F_2$, 对应两个竞争风险时的比率 $\mathrm{电阻}\left(吨\right)\equiv F_1\left(吨\right)∕F_2\left(吨\right)$ 在不减少 $吨>0$. 首先，我们在此顺序约束下的连续情况下推导出这些 CIF 的非参数最大似然估计量 (NPMLE)，并表明它们是不一致的。然后，我们开发一致强一致的投影类型估计器，并研究结果过程的弱收敛。通过模拟，我们比较了 NPMLE 和我们的估计器的有限样本性能，并表明我们的估计器在我们考虑的所有场景中的均方误差方面总体上优于它们。我们还开发了一个测试这个顺序约束的存在，并将所有这些结果扩展到审查案例。为了说明我们开发的理论的适用性，我们提供了一个现实生活中的例子。