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In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this article, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is the sphere and the cost function is the distance squared. As an application to statistics, a new family of probability densities on the sphere is defined in terms of cost-convex functions. The log-concave property of the likelihood function follows from the inequality. [ABSTRACT FROM AUTHOR]

Abstract: We study properties of Fisher distribution (von Mises–Fisher distribution, matrix Langevin distribution) on the rotation group . In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011) [16], and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data. [Copyright &y& Elsevier]

Abstract: We give a new algorithm to find local maxima and minima of a holonomic function and apply it for the Fisher–Bingham integral on the sphere , which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems. [Copyright &y& Elsevier]