{"title":"Generalization of the Alpha-Stable Distribution with the Degree of Freedom","authors":"Stephen H. Lihn","doi":"arxiv-2405.04693","DOIUrl":null,"url":null,"abstract":"A Wright function based framework is proposed to combine and extend several\ndistribution families. The $\\alpha$-stable distribution is generalized by\nadding the degree of freedom parameter. The PDF of this two-sided super\ndistribution family subsumes those of the original $\\alpha$-stable, Student's t\ndistributions, as well as the exponential power distribution and the modified\nBessel function of the second kind. Its CDF leads to a fractional extension of\nthe Gauss hypergeometric function. The degree of freedom makes possible for\nvalid variance, skewness, and kurtosis, just like Student's t. The original\n$\\alpha$-stable distribution is viewed as having one degree of freedom, that\nexplains why it lacks most of the moments. A skew-Gaussian kernel is derived\nfrom the characteristic function of the $\\alpha$-stable law, which maximally\npreserves the law in the new framework. To facilitate such framework, the\nstable count distribution is generalized as the fractional extension of the\ngeneralized gamma distribution. It provides rich subordination capabilities,\none of which is the fractional $\\chi$ distribution that supplies the needed\n'degree of freedom' parameter. Hence, the \"new\" $\\alpha$-stable distribution is\na \"ratio distribution\" of the skew-Gaussian kernel and the fractional $\\chi$\ndistribution. Mathematically, it is a new form of higher transcendental\nfunction under the Wright function family. Last, the new univariate symmetric\ndistribution is extended to the multivariate elliptical distribution\nsuccessfully.","PeriodicalId":501139,"journal":{"name":"arXiv - QuantFin - Statistical Finance","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - QuantFin - Statistical Finance","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.04693","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
A Wright function based framework is proposed to combine and extend several
distribution families. The $\alpha$-stable distribution is generalized by
adding the degree of freedom parameter. The PDF of this two-sided super
distribution family subsumes those of the original $\alpha$-stable, Student's t
distributions, as well as the exponential power distribution and the modified
Bessel function of the second kind. Its CDF leads to a fractional extension of
the Gauss hypergeometric function. The degree of freedom makes possible for
valid variance, skewness, and kurtosis, just like Student's t. The original
$\alpha$-stable distribution is viewed as having one degree of freedom, that
explains why it lacks most of the moments. A skew-Gaussian kernel is derived
from the characteristic function of the $\alpha$-stable law, which maximally
preserves the law in the new framework. To facilitate such framework, the
stable count distribution is generalized as the fractional extension of the
generalized gamma distribution. It provides rich subordination capabilities,
one of which is the fractional $\chi$ distribution that supplies the needed
'degree of freedom' parameter. Hence, the "new" $\alpha$-stable distribution is
a "ratio distribution" of the skew-Gaussian kernel and the fractional $\chi$
distribution. Mathematically, it is a new form of higher transcendental
function under the Wright function family. Last, the new univariate symmetric
distribution is extended to the multivariate elliptical distribution
successfully.