Editing Stable distribution (section)

== Simulation of stable variates ==
There are no analytic expressions for the inverse <math>F^{-1}(x)</math> nor the CDF <math>F(x)</math> itself, so the inversion method cannot be used to generate stable-distributed variates.<ref name ="Nolan 1997">{{Cite journal|title = Numerical calculation of stable densities and distribution functions |journal = Communications in Statistics. Stochastic Models| date = 1997 | issn = 0882-0287 |pages = 759–774 | volume = 13 |issue = 4| doi = 10.1080/15326349708807450 | first = John P. | last = Nolan}}</ref><ref name=":4">{{Cite journal|last=Lihn|first=Stephen| date=2017| title=A Theory of Asset Return and Volatility Under Stable Law and Stable Lambda Distribution| url=https://ssrn.com/abstract=3046732| journal=SSRN}}</ref> Other standard approaches like the rejection method would require tedious computations. An elegant and efficient solution was proposed by Chambers, Mallows and Stuck (CMS),<ref>{{Cite journal|title = A Method for Simulating Stable Random Variables|journal = Journal of the American Statistical Association|date = 1976|issn = 0162-1459| pages = 340–344| volume = 71 |issue = 354| doi = 10.1080/01621459.1976.10480344 |first1 = J. M.|last1 = Chambers|first2 = C. L.|last2 = Mallows|first3 = B. W.|last3 = Stuck}}</ref> who noticed that a certain integral formula<ref>{{Cite book|title = One-Dimensional Stable Distributions|last = Zolotarev | first = V. M.|publisher = American Mathematical Society|year = 1986|isbn = 978-0-8218-4519-6|url-access = registration|url = https://archive.org/details/onedimensionalst00zolo_0}}</ref> yielded the following algorithm:<ref>{{Cite book | title = Heavy-Tailed Distributions in VaR Calculations|publisher = Springer Berlin Heidelberg| date = 2012|isbn = 978-3-642-21550-6|pages = 1025–1059|series = Springer Handbooks of Computational Statistics | doi = 10.1007/978-3-642-21551-3_34 |first1 = Adam | last1 = Misiorek|first2 = Rafał|last2 = Weron | url = http://www.im.pwr.wroc.pl/~hugo/RePEc/wuu/wpaper/HSC_10_05.pdf | editor-first = James E.|editor-last = Gentle | editor-first2 = Wolfgang Karl| editor-last2 = Härdle| editor-first3 = Yuichi |editor-last3 = Mori}}</ref>
* generate a random variable <math>U</math> uniformly distributed on <math>\left (-\tfrac{\pi}{2},\tfrac{\pi}{2} \right )</math> and an independent [[Exponential_distribution#Random_variate_generation|exponential random variable]] <math>W</math> with mean 1;
* for <math>\alpha\ne 1</math> compute: <math display="block">X = \left (1+\zeta^2 \right )^\frac{1}{2\alpha} \frac{\sin ( \alpha(U+\xi)) }{ (\cos(U))^{\frac{1}{\alpha}}} \left (\frac{\cos (U - \alpha(U+\xi)) }{W} \right )^\frac{1-\alpha}{\alpha},</math>
* for <math>\alpha=1</math> compute: <math display="block">X = \frac{1}{\xi}\left\{\left(\frac{\pi}{2}+\beta U \right)\tan U- \beta\log\left(\frac{\frac{\pi}{2} W\cos U}{\frac{\pi}{2}+\beta U}\right)\right\},</math> where <math display="block">\zeta = -\beta\tan\frac{\pi\alpha}{2}, \qquad \xi =\begin{cases}
\frac{1}{\alpha} \arctan(-\zeta) & \alpha \ne 1 \\
\frac{\pi}{2} & \alpha=1
\end{cases}</math>

This algorithm yields a random variable <math>X\sim S_\alpha(\beta,1,0)</math>. For a detailed proof see.<ref>{{Cite journal| title = On the Chambers-Mallows-Stuck method for simulating skewed stable random variables| journal = Statistics & Probability Letters|date = 1996|pages = 165–171 |volume = 28|issue = 2|doi = 10.1016/0167-7152(95)00113-1| first = Rafał |last = Weron | citeseerx = 10.1.1.46.3280| s2cid = 9500064}}</ref>

To simulate a stable random variable for all admissible values of the parameters <math>\alpha</math>, <math>c</math>, <math>\beta</math> and <math>\mu</math> use the following property: If <math>X \sim S_\alpha(\beta,1,0)</math> then
<math display="block">Y = \begin{cases}
c X+\mu & \alpha \ne 1 \\
c X+\frac{2}{\pi}\beta c\log c + \mu & \alpha = 1
\end{cases}</math>
is <math>S_\alpha(\beta,c,\mu)</math>. For <math>\alpha = 2</math> (and <math>\beta = 0</math>) the CMS method reduces to the well known [[Box–Muller transform|Box-Muller transform]] for generating [[Normal distribution|Gaussian]] random variables.<ref>{{Cite book|title = Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes|last1 = Janicki |first1 = Aleksander | publisher = CRC Press| year = 1994|isbn = 9780824788827 | url = https://www.crcpress.com/Simulation-and-Chaotic-Behavior-of-Alpha-stable-Stochastic-Processes/Janicki-Weron/9780824788827 | last2 = Weron | first2 = Aleksander}}</ref> While other approaches have been proposed in the literature, including application of Bergström<ref>{{Cite journal|title = Fast, accurate algorithm for numerical simulation of Lévy stable stochastic processes|journal = Physical Review E|date = 1994|pages = 4677–4683| volume = 49|issue = 5| doi = 10.1103/PhysRevE.49.4677| first = Rosario Nunzio |last = Mantegna | pmid = 9961762| bibcode = 1994PhRvE..49.4677M}}</ref> and LePage<ref>{{Cite journal|title = Computer investigation of the Rate of Convergence of Lepage Type Series to α-Stable Random Variables|journal = Statistics| date = 1992|issn = 0233-1888| pages = 365–373|volume = 23|issue = 4|doi = 10.1080/02331889208802383|first1 = Aleksander|last1 = Janicki|first2 = Piotr| last2 = Kokoszka}}</ref> series expansions, the CMS method is regarded as the fastest and the most accurate.