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) , a skewness parameter , a scale parameter , and a location parameter . The first two are shape parameters. Two possible parametrizations include

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

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

Stable distributions are implemented in the Wolfram Language as StableDistribution[type, alpha, beta, mu, sigma], where type is 0 or 1 and specifies the parametrization, alpha is the index of stability, beta is the skewness parameter, mu is the location parameter, and sigma is the scale parameter.

References

Lévy, P. Calcul des probabilités. Paris, France: Gauthier-Villars, 1925.

Rimmer, R. H. and Nolan, J. P. "Stable Distributions in Mathematica." Mathematica J. 9, 776-789, 2005.

See also