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New Fréchet features for random distributions and associated sensitivity indices

In this article we define new FrÈchet features for random cumulative distribution functions using contrast. These contrasts allow to construct Wasserstein costs and our new features minimize the average costs as the FrÈchet mean minimizes the mean square Wasserstein$_2$ distance. An example of new features is the median, and more generally the quantiles. From these definitions, we are able to define sensitivity indices when the random distribution is the output of a stochastic code. Associated to the FrÈchet mean we extend the Sobol indices, and in general the indices associated to a contrast that we previously proposed.