Matrix-variate statistical distributions and fractional calculus

Abstract

Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional differential equations, Mittag-Leffler functions and Fox Hfunction appear naturally. Some results connected with generalized MittagLeffler density and their asymptotic behavior will be considered. Reference is made to applications in physics, particularly superstatistics and nonextensive statistical mechanics.MSC 2010 : 15A15, 15A52, 33C60, 33E12, 44A20, 62E15

Hypergeometric Functions: From One Scalar Variable to Several Matrix Arguments, in Statistics and Beyond

“…In Theoretical Statistics, several recent research results on hypergeometric functions use fractional calculus (Mathai [44]), associated with functions of matrix arguments (Mathai and Haubold [62]). But it is still too early to appraise the impact of this notion on Statistics.…”

“…We have already mentioned the works of James [13] [62]), most of them still at the very theoretical level, however. They will probably make an impact on statistics in the years ahead.…”

On the Multi-Index Mittag-Leffler Functions and Their Mellin Transforms

“…Recently the interest to M-L type functions and their generalizations has grown up in view of their important role in fractional calculus and related integral and differential equations of fractional order (as their solutions) and applications [35], for example in modelling some evolution problems [7], fractional diffusion processes [24], nonlinear waves, etc. The M-L function enjoys also applications in the stochastic processes, statistical distributions and conditional expectations as the so-called M-L (probability) density, see for example Mathai, Haubold et al [27], [8], [9], etc.…”

On the multi-index Mittag-Leffler series and their boundary behaviour

“…The M-L functions are also used in the statistical distributions and conditional expectations, usually known as M-L statistical density, see, for example, Mathai-Haubold. [14] In 1971, Prabhakar [15] generalized (1.1) by introducing the 3-parametric function…”