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Stable Distributions in Mathematica
Robert H. Rimmer
John P. Nolan

Parameterizations of Stable Laws

A general stable distribution requires four parameters to describe it:

  • index of stability or characteristic exponent
  • skewness parameter
  • scale parameter
  • location parameter

There are multiple parameterizations for stable laws, and much confusion has been caused by these different parameterizations. This package includes two parameterizations. The characteristic functions in the and parameterizations follow.

:

and

:

The parameterization is the best choice for numerical computation of stable distributions, since it has the simplest form for the characteristic function that is continuous in all four parameters. The parameterization has been commonly used in economic literature and has the property that the location parameter, Delta, is the mean when . The following graphics array demonstrates these two parameterizations. The scale parameter, Gamma, has been set to 1 and the location parameter, Delta, is 0. Each graph shows varying values of Beta ( in black, red, green, and blue colors, respectively) at the value of Alpha and parameterization type listed on the graph. Negative values of Beta would give mirror image graphs. When Alpha is near 1, the mode in the parameterization moves dramatically; the parameterization progressively reduces this motion as Alpha approaches 1.



     
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