Risk Estimation under Probabilistic Symmetries, 13 March 2017

Enrico Ferri, Carlos Vázquez, José Luis Fernández: Consistency and qualitative robustness, two of the main forms of stability usually required when dealing with risk estimators, are presented in an overall perspective by considering different notions of probabilistic symmetries at the level of the stochastic drivers. For such a purpose, a general topological framework is properly introduced by extending the common notion of the weak topology of measures. In particular, the main objective of the work is to state all the results in terms of exchangeability, the celebrated notion of distributional invariance arising when dealing with arbitrary permutations of finite order. In this respect, a new concept of qualitative robustness is defined in terms of the perturbations that affect the system when data are slightly perturbed, and a refined version of the celebrated Hampel’s theorem is provided. Such an overture turns out to be strongly appealing from the conceptual point of view, since data are the main drivers of the empirical analysis.