Model averaging in the space of probability distributions

This work investigates the problem of model averaging in the context of measure-valued data. Specifically, we study aggregation schemes in the space of probability distributions metrized in terms of the Wasserstein distance. The resulting aggregate models, defined via Wasserstein barycenters, are optimally calibrated to empirical data. To enhance model performance, we employ regularization schemes motivated by the standard elastic net penalization, which is shown to consistently yield models enjoying sparsity properties. The consistency properties of the proposed averaging schemes with respect to sample size are rigorously established using the variational framework of $\Gamma$-convergence. The performance of the methods is evaluated through carefully designed synthetic experiments that assess behavior across a range of distributional characteristics and stress conditions. Finally, the proposed approach is applied to a real-world dataset of insurance losses - characterized by heavy-tailed behavior - to estimate the claim size distribution and the associated tail risk.