Let $F(x)$ be a probability distribution function. Assuming the singular part to be identically zero, it is well known (see e.g. Cramer  pp. 52, 53) that $F(x)$ can be decomposed into $F(x) = F_1(x) + F_2(x)$ where $F_1(x)$ is an everywhere continuous function and $F_2(x)$ is a pure step function with steps of magnitude, say, $S_\nu$ at the points $x = x_\nu, \nu = 1, 2, \cdots, \infty$ and that finally both $F_1(x)$ and $F_2(x)$ are non-decreasing and are uniquely determined. In this paper the problem of estimating the jump $S_i$ corresponding to the saltus $x = x_i$ is considered. Also considered are the problems of estimation of reliability and hazard rate. Based on a random sample $X_1, X_2, \cdots X_n$ of size $n$ from the distribution $F(x)$, consistent and asymptotically normal classes of estimators are obtained for estimating the jump $S_i$ corresponding to the saltus $x = x_i$. Based on the earlier work of the author  on estimation of probability density, consistent and asymptotically normal estimates are obtained for the reliability and hazard rate.
"Estimation of Jumps, Reliability and Hazard Rate." Ann. Math. Statist. 36 (3) 1032 - 1040, June, 1965. https://doi.org/10.1214/aoms/1177700075