Applications of fractional calculus to spherical (radial) probability models and generalizations

Abstract

In the statistical study of radial (known also as spherical) probability models, conditional distributions ("Thompson distributions") on the sphere and minimum variance unbiased estimation involve fractional calculus. Projections of spherical distributions on lower dimension spaces, that is, distributions of projections of spherically distributed vectors and their "anti-projections"—which generalize univariate symmetric models to multivariate models—are obtained by fractional integration and differentiation. A calculus of operators based on the group of Fourier-Hankel type transforms and fractional integration-differentiation to apply to probability densities of spherical models is outlined. The problem of comparing spherical models to the normal Gaussian model through series expansion is touched upon. A matric generalization of fractional calculus, motivated by the study of the family of hyperspherical distributions of random matrices, is suggested for further research and consideration.