Shape of a Distribution Through the L2-Wasserstein Distance

Abstract Let Q be a probability measure on ℝ d and let ℑ be a family of probability measures on ℝ d which will be considered as a pattern. For suitable patterns we consider the closest law to Q in ℑ, through the L2-Wasserstein distance, as a descriptive measure associated to Q. The distance between Q and ℑ is a natural measure of the fit of Q to the pattern.