Energy distances for statistical inference on infinite dimensional Hilbert spaces without moment conditions

For statistical inference on an infinite-dimensional Hilbert space H with no moment conditions we introduce a new class of energy distances on the space of probability measures on H. The proposed distances consist of the integrated squared modulus of the corresponding difference of the characteristic functionals with respect to a reference probability measure on the Hilbert space. Necessary and sufficient conditions are established for the reference probability measure to be characteristic, the property that guarantees that the distance defines a metric on the space of probability measures on H. We also use these results to define new distance covariances, which can be used to measure the dependence between the marginals of a two dimensional distribution of H2 without existing moments. On the basis of the new distances we develop statistical inference for Hilbert space valued data, which does require any moment assumptions. As a consequence, our methods are robust with respect to heavy tails in finite dimensional data. In particular, we consider the problem of comparing the distributions of two samples and the problem of testing for independence and construct new minimax optimal tests for the corresponding hypotheses. We also develop aggregated (with respect to the reference measure) procedures for power enhancement and investigate the finite-sample properties by means of a simulation study.