The paper is devoted to a recently introduced class of nonlinear digital filters, namely the stack filters which includes rank order filters, standard median filters, some morphological ones, etc. The particularity of such a class lies in the fact that it is related to the statistical properties of the input signal. The exposition is based on three new concepts: breakdown points, breakdown probabilities and midpoint sensitivity curves. The developed theory meets the requirements related to optimization, with those related to imposed constraints. Numerical examples are given. We think the paper represents a basic contribution in the field of stack filters and will be of high interest to engineers working in this area.