A Deformed Exponential Statistical Manifold

Consider [Formula: see text] a probability measure and [Formula: see text] the set of [Formula: see text]-equivalent strictly positive probability densities. To endow [Formula: see text] with a structure of a [Formula: see text]-Banach manifold we use the [Formula: see text]-connection by an open arc, where [Formula: see text] is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and [Formula: see text]-exponential functions. Moreover, we find the tangent space of [Formula: see text] at a point p, and as a consequence the tangent bundle of [Formula: see text]. We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and [Formula: see text]-exponential functions can be used to generalize of Rényi divergence.