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Shape of a distribution through the $$L_2$$-Wasserstein distance. (English) Zbl 1135.62333
Cuadras, Carles M. (ed.) et al., Distributions with given marginals and statistical modelling. Papers presented at the meeting, Barcelona, Spain, July 17–20, 2000. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0914-3/hbk). 51-61 (2002).
Summary: Let $$Q$$ be a probability measure on $$\mathbb R^d$$ and let $$\operatorname{Im}$$ be a family of probability measures on $$\mathbb R^d$$ which will be considered as patterns. For suitable patterns we consider the closest law to $$Q$$ in $$\operatorname{Im}$$, through the $$L_2$$-Wasserstein distance, as a descriptive measure associated to $$Q$$. The distance between $$Q$$ and $$\operatorname{Im}$$ is a natural measure of the fit of $$Q$$ to the pattern. We analyze this approach via the consideration of different patterns. Some of them generalize usual location and dispersion measures. Special attention will be paid to patterns based on uniform distributions on suitable families of sets, like the intervals in the unidimensional case (which allows us to analyze the flatness of the one-dimensional distributions) or the ellipsoids for the multivariate distributions.
For the entire collection see [Zbl 1054.62002].

##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62E10 Characterization and structure theory of statistical distributions
##### Keywords:
flatness; shape of distributions