Some invariant procedures, which are essentially step-functions, are considered as estimators of the cumulative distribution function of a one-dimensional random variable on which a finite fixed number of observations are given, for various loss functions. Two principal classes of loss functions are considered and it is shown that for a special loss function in one class the optimum procedure is the usual sample cumulative function.
Om P. Aggarwal. "Some Minimax Invariant Procedures for Estimating a Cumulative Distribution Function." Ann. Math. Statist. 26 (3) 450 - 463, September, 1955. https://doi.org/10.1214/aoms/1177728490