Abstract
This paper carries out an investigation of the orthogonal-polynomial approach to reshaping symmetric distributions to fit in with data requirements so as to cover the multivariate case. With this objective in mind, reference is made to the class of spherical distributions, given that they provide a natural multivariate generalization of univariate even densities. After showing how to tailor a spherical distribution via orthogonal polynomials to better comply with kurtosis requirements, we provide operational conditions for the positiveness of the resulting multivariate Gram–Charlier-like expansion, together with its kurtosis range. Finally, the approach proposed here is applied to some selected spherical distributions.
Highlights
It is well known that a set of orthogonal polynomials can be associated to a density function with existing moments of all order
This method—which can be viewed as an inheritance of the Gram–Charlier expansion—has proved effective when applied to distributions to account for possibly severe kurtosis and skewness (e.g., [5,6])
After running through the procedure to derive a spherical distribution via a density generator, the Gram–Charlier-like expansion is duly specified, and the involved orthogonal polynomial derived in terms of the moments of the so-called modular variable
Summary
It is well known that a set of orthogonal polynomials can be associated to a density function with existing moments of all order This clears the way for the tailoring of the shape of a given distribution from “inside” through a polynomial shape-adapter, built on the orthogonal polynomials engendered by the same distribution (e.g., [1,2,3,4]). After running through the procedure to derive a spherical distribution via a density generator, the Gram–Charlier-like expansion is duly specified, and the involved orthogonal polynomial derived in terms of the moments of the so-called modular variable (see Appendix A). An appendix, devoted to spherical laws, completes the paper
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