Information geometry of tempered stable processes

Abstract

We derive the information geometry of tempered stable processes. We first compute the $\alpha$-divergence between two tempered stable processes. From the divergence, we obtain the Fisher information matrix and the $\alpha$-connection on the statistical manifold. The generalized, classical, and rapidly-decreasing tempered stable geometries are dually flat, with potential functions and canonical divergences expressed in terms of the tempering parameters. Moreover, their $\alpha$-curvature tensors vanish for every $\alpha$. The geometric results are also applied to bias reduction in maximum likelihood estimation and Bayesian predictive priors.