A two-parameter family of non-parametric, deformed exponential manifolds

Abstract

AbstractWe construct a new family of non-parametric statistical manifolds by means of a two-parameter class of deformed exponential functions, that includes functions with power-law, linear and sublinear rates of growth. The manifolds are modelled on weighted, mixed-norm Sobolev spaces that are especially suited to this purpose, in the sense that an important class of nonlinear superposition operators (those used in the construction of divergences and tensors) act continuously on them. We analyse variants of these operators, that map into “subordinate” Sobolev spaces, and evaluate the associated gain in regularity. With appropriate choice of parameter values, the manifolds support a large variety of the statistical divergences and entropies appearing in the literature, as well as their associated tensors, eg. the Fisher-Rao metric. Manifolds of finite measures and probability measures are constructed; the latter are shown to be smoothly embedded submanifolds of the former.