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The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

Dipartimento di Discipline matematiche, Finanza matematica ed Econometria, Università Cattolica del Sacro Cuore, Largo Gemelli 1, Milano 20123, Italy
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Academic Editor: Charles F. Dunkl
Symmetry 2016, 8(8), 77; https://doi.org/10.3390/sym8080077
Received: 14 July 2016 / Revised: 28 July 2016 / Accepted: 30 July 2016 / Published: 5 August 2016
(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)
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. View Full-Text
Keywords: orthogonal polynomials; spherical distributions; Gram–Charlier-like expansions; kurtosis orthogonal polynomials; spherical distributions; Gram–Charlier-like expansions; kurtosis
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Bagnato, L.; Faliva, M.; Zoia, M.G. The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements. Symmetry 2016, 8, 77.

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