Abstract
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 px = 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.