A diffeomorphism between statistical manifolds is said to be statistical if it preserves statistical structures. Our purpose is to find conditions that guarantee an extension of a given linear isomorphism between given tangent spaces to a local statistical diffeomorphism. In Riemannian geometry, it is known as the Cartan-Ambrose-Hicks theorem, which implies that a Riemannian metric is locally determined by its Riemannian curvature tensor. We generalize this theorem for statistical manifolds, and, in particular, for Hessian manifolds. We prove that a statistical structure is locally characterized by its Riemannian curvature tensor and difference tensor. Furthermore, we show that a Hessian structure is locally determined by its Hessian curvature tensor and difference tensor.

information geometry, statistical diffeomorphism, statistical manifold, Hessian manifold.