In this paper the minimax estimators of a cumulative distribution function $F$ is obtained for four types of loss functions. The result is quite general in that no restrictions are imposed on the unknown $F$. Moreover, the estimates do not depend upon the weight function used in the definition of the loss functions. It is also shown that the sample distribution function is minimax under one of these types of loss functions.
"Minimax Estimation of a Cumulative Distribution Function." Ann. Statist. 1 (6) 1149 - 1157, November, 1973. https://doi.org/10.1214/aos/1176342563