Minimaxity and equivariance in infinite dimension

In a location model on an infinite-dimensional Banach space, we study the problem of estimating the location parameter. This estimation problem exhibits an invariance structure where the group involved is the Banach space itself. Under suitable conditions it is shown that the minimax risk coincides with the minimum risk over all equivariant estimators, thus establishing the minimaxity of minimum risk equivariant estimators. Furthermore, such estimators are shown to be extended Bayes estimators, and least favorable sequences of prior distributions are derived. The proofs rely on a general result for structure models and a concentration condition for probability measures on a Banach space related to reproducing kernel Hilbert spaces of Gaussian measures.