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Stable Distribution

Stable distributions are a class of probability distributions allowing skewness and heavy tails (Rimmer and Nolan 2005). They are described by an index of stability (also known as a characteristic exponent) alpha in (0,2], and skewness parameter beta in [-1,1], a scale parameter gamma>0, and a location parameter delta in R. Two possible parametrizations include

S_1(alpha,beta,gamma,delta)={exp{iudelta-gamma^alpha|u|^alpha[1+ibeta(|ugamma|^(1-alpha)-1)]sgn(u)tan(1/2pialpha)} for alpha!=1; exp{iudelta-(gamma|u|[pi+2ibetaln(|ugamma|)]sgn(u))/pi} for alpha=1
(1)
S_2(alpha,beta,gamma,delta)={exp{iudelta-gamma^alpha|u|^alpha[1-ibetasgn(u)tan(1/2pialpha)]} for alpha!=1; exp{iudelta-gamma|u|(1+(2ibetaln[|u|]sgn(u))/pi)} for alpha=1
(2)

(Rimmer and Nolan 2005). S_1 is most convenient for numerical computations, whereas S_2 is commonly used in economics.

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References

Lévy, P. Calcul des probabilités. Paris: Gauthier-Villars, 1925.Nolan, J. P. Stable Distributions: Models for Heavy Tailed Data. Boston, MA: Birkhäuser, 2005. Nolan, J. P. "Stable MathLink Package." http://www.robustanalysis.com/.Rimmer, R. H. and Nolan, J. P. "Stable Distributions in Mathematica." Mathematica J. 9, 776-789, 2005.

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Stable Distribution

Cite this as:

Weisstein, Eric W. "Stable Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StableDistribution.html

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