New Fr\'echet features for random distributions and associated sensitivity indices

In this article we define new Fr`Echet 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 Fr`Echet 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 Fr`Echet mean we extend the Sobol indices, and in general the indices associated to a contrast that we previously proposed.