Set Theory

This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject

Detecting stochastic dominance for poset-valued random variables as an example of linear programming on closure systems

In this paper we develop a linear programming method for detecting stochastic dominance for random variables with values in a partially ordered set (poset) based on the upset-characterization of stochastic dominance. The proposed detection-procedure is based on a descriptively interpretable statistic, namely the maximal probability-difference of an upset. We show how our method is related to the general task of maximizing a linear function on a closure system. Since closure systems are describable via their valid formal implications, we can use here ingredients of formal concept analysis. We also address the question of inference via resampling and via conservative bounds given by the application of Vapnik-Chervonenkis theory, which also allows for an adequate pruning of the envisaged closure system that allows for the regularization of the test statistic (by paying a price of less conceptual rigor). We illustrate the developed methods by applying them to a variety of data examples, concretely to multivariate inequality analysis, item impact and differential item functioning in item response theory and to the analysis of distributional differences in spatial statistics. The power of regularization is illustrated with a data example in the context of cognitive diagnosis models