Stable Distribution Function

The simple algorithm used in this Demonstration can calculate the stable distribution function and its first several derivatives with good accuracy for . It is offered to help with financial analysis where the data generally has the shape parameter greater than 1. The Nolan 1-parameterization is used where the parameters have the characteristics listed below.
is the distribution shape parameter, . For the result is the normal distribution; is the tail exponent of the distribution: lower values give fatter tails.
is the skewness parameter in the range (-1, 1).
is the scale parameter.
is the location parameter; when as in this Demonstration is the expectation of the distribution.


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The speed of the algorithm may be considerably enhanced for the density and distribution functions by implementing each derivative separately and using only the real component of the integrand. For example the code samples below give the integrands for the distribution function, followed by the density function for .
NIntegrate in recent versions of Mathematica can handle this integrand quite well. Be sure to set: Method -> {Automatic, "SymbolicProcessing" -> 0}.
For many applications in finance and software for stable distributions visit mathestate.
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