Barycenter and maximum likelihood

We refine recent existence and uniqueness results, for the barycenter of points at infinity of Hadamard manifolds, to measures on the sphere at infinity of symmetric spaces of non compact type and, more specifically, to measures concentrated on single orbits. The barycenter will be interpreted as the maximum likelihood estimate (MLE) of generalized Cauchy distributions on Furstenberg boundaries. As a spin-off, a new proof of the general Knight–Meyer characterization theorem will be given

Angular Gaussian and Cauchy estimation

This paper proposes a unified treatment of maximum likelihood estimates of angular Gaussian and multivariate Cauchy distributions in both the real and the complex case. The complex case is relevant in shape analysis. We describe in full generality the set of maxima of the corresponding log-likelihood functions with respect to an arbitrary probability measure. Our tools are the convexity of log-likelihood functions and their behaviour at infinity