Level Sets Based Distances for Probability Measures and Ensembles with Applications

In this paper we study Probability Measures (PM) from a functional point of view: we show that PMs can be considered as functionals (generalized functions) that belong to some functional space endowed with an inner product. This approach allows us to introduce a new family of distances for PMs, based on the action of the PM functionals on `interesting' functions of the sample. We propose a specific (non parametric) metric for PMs belonging to this class, based on the estimation of density level sets. Some real and simulated data sets are used to measure the performance of the proposed distance against a battery of distances widely used in Statistics and related areas.