Estimation of the $k$th derivative of a df by means of the $k$th-order difference quotients of the empiric df is investigated. In particular, consistency conditions are given, the asymptotic bias, variance, and mean-squared error of the estimator are computed, and means of minimizing the latter are discussed.
"Estimation of the $k$th Derivative of a Distribution Function." Ann. Statist. 2 (2) 359 - 361, March, 1974. https://doi.org/10.1214/aos/1176342670