Two new properties of the kernel distribution function estimator of diverse nature are derived. Firstly, a law of the iterated logarithm is proved for both the integrated absolute error and the integrated squared error of the estimator. Secondly, the maximal smoothing principle in kernel density estimation developed by Terrell is extended to kernel distribution function estimation, which allows, among others, the derivation of an alternative quick-and-simple bandwidth selector. In fact, there is a common link between the two topics: both problems are solved through the use of the same, not-so-standard, methodology. The results based on simulated data and a real data set are also presented.

推导了具有不同性质的核分布函数估计器的两个新属性。首先，针对估计器的积分绝对误差和积分平方误差，证明了迭代对数定律。其次，由Terrell开发的内核密度估计中的最大平滑原理扩展到了内核分布函数估计，这除其他事项外，还允许推导另一种快速简便的带宽选择器。实际上，这两个主题之间存在一个共同的纽带：这两个问题都是通过使用相同的，不太标准的方法来解决的。还提供了基于模拟数据和真实数据集的结果。