Exact estimations of empirical risk bias for discrete feature

The work is devoted to a problem of statistical robustness of deciding functions, or risk estimation. By risk we mean some measure of decision function prediction quality, for example, an error probability. For the case of discrete "independent" variable the dependence of average risk on empirical risk for the "worst" distribution ("strategies of nature") is obtained. The result gives exact value of empirical risk bias that allows evaluating an accuracy of Vapnik-Chervonenkis risk estimations. To find a distribution providing maximum of empirical risk bias one need to solve an optimization problem on function space. The problem being very complicate in general case appears to be solvable when the "independent" feature is a space of isolated points. The space has low practical use but it allows scaling well-known estimations by Vapnik and Chervonenkis. A heuristic approach for using the obtained results for estimating a quality of deciding functions in general case (multidimensional space of discrete and continuous features) is also suggested.