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第198行：
第198行： − − − − + + + + + + + + + − + + + + + − *the [[Log-normal distribution]]; − *the [[Lévy distribution]]; − *the [[Weibull distribution]] with shape parameter greater than 0 but less than 1;
→Common heavy-tailed distributions
==Common heavy-tailed distributions==
==Common heavy-tailed distributions==
All commonly used heavy-tailed distributions are subexponential.<ref name="Embrechts"/>
All commonly used heavy-tailed distributions are subexponential.<ref name="Embrechts"/>
Those that are one-tailed include:
Those that are one-tailed include:
*the [[Pareto distribution]];
*the [[Log-normal distribution]];
*the [[Lévy distribution]];
*the [[Weibull distribution]] with shape parameter greater than 0 but less than 1;
*the [[Burr distribution]];
*the [[log-logistic distribution]];
*the [[log-gamma distribution]];
*the [[Fréchet distribution]];
*the [[log-Cauchy distribution]], sometimes described as having a "super-heavy tail" because it exhibits [[logarithmic growth|logarithmic decay]] producing a heavier tail than the Pareto distribution.<ref>{{cite book|title=Laws of Small Numbers: Extremes and Rare Events|author=Falk, M., Hüsler, J. & Reiss, R.|page=80|year=2010|publisher=Springer|isbn=978-3-0348-0008-2}}</ref><ref>{{cite web|title=Statistical inference for heavy and super-heavy tailed distributions|url=http://docentes.deio.fc.ul.pt/fragaalves/SuperHeavy.pdf|author=Alves, M.I.F., de Haan, L. & Neves, C.|date=March 10, 2006|access-date=November 1, 2011|archive-url=https://web.archive.org/web/20070623175435/http://docentes.deio.fc.ul.pt/fragaalves/SuperHeavy.pdf|archive-date=June 23, 2007|url-status=dead}}</ref>
*the [[Pareto distribution]];
Those that are two-tailed include:
*The [[Cauchy distribution]], itself a special case of both the stable distribution and the t-distribution;
*The family of [[stable distributions]],<ref>{{cite web |author=John P. Nolan | title=Stable Distributions: Models for Heavy Tailed Data| year=2009 | url=http://academic2.american.edu/~jpnolan/stable/chap1.pdf | accessdate=2009-02-21}}</ref> excepting the special case of the normal distribution within that family. Some stable distributions are one-sided (or supported by a half-line), see e.g. [[Lévy distribution]]. See also ''[[financial models with long-tailed distributions and volatility clustering]]''.
*The [[Student's t-distribution|t-distribution]].
*The skew lognormal cascade distribution.<ref>{{cite web | author=Stephen Lihn | title=Skew Lognormal Cascade Distribution | year=2009 | url=http://www.skew-lognormal-cascade-distribution.org/ | access-date=2009-06-12 | archive-url=https://web.archive.org/web/20140407075213/http://www.skew-lognormal-cascade-distribution.org/ | archive-date=2014-04-07 | url-status=dead }}</ref>
Category:Tails of probability distributions
Category:Tails of probability distributions