Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X-1,..., X-n) be a sample from an unknown cumulative distribution function F defined on the real line R. The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t; X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situationwhen the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.