We consider-135 nonparametric estimation of a distribution function F associated with a random variable X based on a random sample. First, for x, a point of continuity of F, we define a class of estimators for F(x) and obtain their rates of convergence. Contrary to the existing literature, we impose no restriction on the existence or smoothness of the derivatives of F. The traditional kernel estimator for F(x) is a member of the class. Second, for x that is either the location of a jump discontinuity or an isolated point of the support, we define a class of estimators for the jump p_x = F(x) − F(x₋) and obtain their rates of convergence. Again, no additional restriction is imposed on F beyond right-continuity. Our results are of significant practical use, as there are numerous examples in economics, finance and biomedicine of distributions that have point masses. Our main insight is also applied to obtain new inversion theorems for characteristic functions and explicit estimates for convergence rates. A small simulation study provides some evidence on the finite sample properties of our proposed estimators and contrasts their performance with some existing alternatives. An empirical section illustrates the use of our estimators using data on global elevation and data associated with “P-hacking” in economics journals.

中文翻译：

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无限制分布及其跳跃的非参数估计

我们考虑基于随机样本的与随机变量X相关的分布函数F的-135 非参数估计。首先，对于F的一个连续点x ，我们为F ( x )定义一类估计量并获得它们的收敛率。与现有文献相反，我们对F的导数的存在性或平滑性没有任何限制。F ( x )的传统核估计器是该类的成员。二、对于x这要么是跳跃不连续的位置，要么是支撑的孤立点，我们为跳跃定义了一类估计量p _x  =  F ( x ) -  F ( x _- ) 并获得它们的收敛率。同样，没有对F施加额外的限制超越正确的连续性。我们的结果具有重要的实际用途，因为在经济学、金融学和生物医学中有许多具有点质量分布的例子。我们的主要见解也适用于获得特征函数的新反演定理和收敛速度的显式估计。一项小型模拟研究为我们提出的估计器的有限样本属性提供了一些证据，并将它们的性能与一些现有的替代方案进行了对比。实证部分说明了我们使用全球海拔数据和经济学期刊中与“ P -hacking”相关的数据的估计器的使用。