  We study the class on non-parametric deformed statistical models where the
deformed exponential has linear growth at infinity and is sub-exponential at
zero. This class generalizes the class introduced by N.J.~Newton. We discuss
the convexity and regularity of the normalization operator, the form of the
deformed statistical divergences and their convex duality, the properties of
the escort densities, and the affine manifold structure of the statistical
bundle.
