Abstract
Let X = (X 1,...,X n ) be a sample from an unknown cumulative distribution function F defined on the real line \(\mathbb{R}\). 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 situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.
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Jokiel-Rokita, A., Magiera, R. Minimax estimation of a cumulative distribution function by converting to a parametric problem. Metrika 66, 61–73 (2007). https://doi.org/10.1007/s00184-006-0094-3
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DOI: https://doi.org/10.1007/s00184-006-0094-3