{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T03:26:01Z","timestamp":1761017161827,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2016,8,5]],"date-time":"2016-08-05T00:00:00Z","timestamp":1470355200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2013Charlier-like expansion, together with its kurtosis range. Finally, the approach proposed here is applied to some selected spherical distributions.<\/jats:p>","DOI":"10.3390\/sym8080077","type":"journal-article","created":{"date-parts":[[2016,8,5]],"date-time":"2016-08-05T10:10:24Z","timestamp":1470391824000},"page":"77","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3129-5385","authenticated-orcid":false,"given":"Luca","family":"Bagnato","sequence":"first","affiliation":[{"name":"Dipartimento di Discipline matematiche, Finanza matematica ed Econometria, Universit\u00e0 Cattolica del Sacro Cuore, Largo Gemelli 1, Milano 20123, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7882-1111","authenticated-orcid":false,"given":"Mario","family":"Faliva","sequence":"additional","affiliation":[{"name":"Dipartimento di Discipline matematiche, Finanza matematica ed Econometria, Universit\u00e0 Cattolica del Sacro Cuore, Largo Gemelli 1, Milano 20123, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8169-781X","authenticated-orcid":false,"given":"Maria","family":"Zoia","sequence":"additional","affiliation":[{"name":"Dipartimento di Discipline matematiche, Finanza matematica ed Econometria, Universit\u00e0 Cattolica del Sacro Cuore, Largo Gemelli 1, Milano 20123, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2016,8,5]]},"reference":[{"key":"ref_1","first-page":"727","article-title":"Moment-based density approximants","volume":"9","author":"Provost","year":"2011","journal-title":"Math. 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Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1007\/s00362-014-0633-3","article-title":"The role of orthogonal polynomials in adjusting hyperbolic secant and logistic distributions to analyse financial asset returns","volume":"56","author":"Bagnato","year":"2015","journal-title":"Stat. Pap."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1080\/03610926.2013.818698","article-title":"Orthogonal polynomials for tailoring density functions to excess kurtosis, asymmetry and dependence","volume":"45","author":"Faliva","year":"2016","journal-title":"Commun. Stat. Theory Methods,"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/0047-259X(81)90082-8","article-title":"On the theory of elliptically contoured distributions","volume":"11","author":"Cambanis","year":"1981","journal-title":"J. Multivar. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"345","DOI":"10.5209\/rev_REMA.2003.v16.n1.16889","article-title":"A survey on continuous elliptical vector distributions","volume":"16","author":"Gomez","year":"2003","journal-title":"Rev. Mat. Complut."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"191","DOI":"10.2307\/3314747","article-title":"A neighborhood-based classifier for LANDSAT data","volume":"12","author":"Owen","year":"1984","journal-title":"Can. J. Stat."},{"key":"ref_10","unstructured":"Szego, G.P. (2004). Risk Measures for the 21st Century, Wiley."},{"key":"ref_11","unstructured":"Chihara, T.S. (1978). An Introduction to Orthogonal Polynomials, Gordon & Breach."},{"key":"ref_12","unstructured":"Bernardo, J.M., DeGroot, M.H., Lindley, D.V., and Smith, A.F.M. (1985). Bayesian Statistics 2, Elsevier and Valencia University Press."},{"key":"ref_13","first-page":"419","article-title":"Distribution theory of Spherical distributions and location-scale parameter generalization","volume":"32","author":"Kelker","year":"1970","journal-title":"Sankhya"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1088\/1469-7688\/2\/4\/201","article-title":"Semi-parametric modelling in finance: theoretical foundations","volume":"2","author":"Bingham","year":"2002","journal-title":"Quant. Financ."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Fang, K., Kotz, S., and Ng, K.W. (1990). Symmetric Multivariate and Related Distributions, Chapman & Hall CRC.","DOI":"10.1007\/978-1-4899-2937-2"},{"key":"ref_16","unstructured":"Gradstein, I., and Ryzhik, I. (1980). Table of Integrals, Series, and Products, Accademic Press."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Oberhettinger, F. (1974). Tables of Mellin Transform, Springer Verlag.","DOI":"10.1007\/978-3-642-65975-1"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/8\/77\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:27:52Z","timestamp":1760210872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/8\/77"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,5]]},"references-count":17,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2016,8]]}},"alternative-id":["sym8080077"],"URL":"https:\/\/doi.org\/10.3390\/sym8080077","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2016,8,5]]}}}