  We develop a general theory to address a consensus-based combination of
estimations in a parallelized or distributed estimation setting. Taking into
account the possibility of very discrepant estimations, instead of a full
consensus we consider a "wide consensus" procedure. The approach is based on
the consideration of trimmed barycenters in the Wasserstein space of
probability distributions on R^d with finite second order moments. We include
general existence and consistency results as well as characterizations of
barycenters of probabilities that belong to (non necessarily elliptical)
location and scatter familes. On these families, the effective computation of
barycenters and distances can be addressed through a consistent iterative
algorithm. Since, once a shape has been chosen, these computations just depend
on the locations and scatters, the theory can be applied to cover with great
generality a wide consensus approach for location and scatter estimation or for
obtaining confidence regions.
