Analytical-Numeric Formulas for the Probability Density Function of Multivariate Stable and Geo-Stable Distributions

Journal of Statistical Theory and Practice, Volume 8, Issue 2, 2014

The main problem in applying the multivariate stable and geo-stable distributions is the lack of a closed-form formula for the probability density function. In this article, we provide a solution by using the connection between these two distributions and fractional calculus. The analytical-numeric formulas for the probability density function for these two probability distributions are obtained via three analytic approximation methods—homotopy perturbation method, Adomian decomposition method, and variational iteration method. Finally, the convergent analysis of the applied methods is discussed.

Keywords: Adomian decomposition method, Fractional calculus, Homotopy perturbation method, Multivariate geo-stable distributions, Multivariate stable distributions, Variational iteration method