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第138行： − + − − All commonly used heavy-tailed distributions are subexponential.<ref name="Embrechts"/> − − 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> − − 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> − − − All commonly used heavy-tailed distributions are subexponential.[6] 第182行：
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→Common heavy-tailed distributions 常见的重尾分布
所有次指数分布都是长尾分布，但可以构造出非次指数分布的长尾分布的示例。
所有次指数分布都是长尾分布，但可以构造出非次指数分布的长尾分布的示例。
== Common heavy-tailed distributions 常见的重尾分布 ==
== 常见的重尾分布 ==
所有常用的重尾分布都是次指数的。<ref name="Embrechts"/>
所有常用的重尾分布都是次指数的。<ref name="Embrechts"/>
* t分布
* t分布
*<font color="#ff8000">偏对数正态级联分布 The skew lognormal cascade distribution</font>。<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>
*<font color="#ff8000">偏对数正态级联分布 The skew lognormal cascade distribution</font>。<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>
== Relationship to fat-tailed distributions 与胖尾分布的关系 ==
== Relationship to fat-tailed distributions 与胖尾分布的关系 ==