Introduction
The Stable MathLink Package is available from John Nolan for $200: http://www.robustanalysis.com.
Free Mathematica 6.0 software.
Stable distributions are a rich class of probability distributions
that allow skewness and heavy tails and have
many intriguing mathematical properties. The class was characterized
by Paul Levy in his study of sums of independent identically distributed
terms in the 1920’s. The lack of closed formulas for
densities and distribution functions for all but a few stable distributions,
Gaussian, Cauchy and Levy, has been
a major drawback to the use of stable distributions by practitioners. There
are now reliable computer programs to compute stable densities, distribution
functions and quantiles. This notebook
demonstrates a MathLink interface to
these programs that makes Mathematica a powerful tool for stable analysis.
Stable distributions have been proposed as a model for many types of physical
and economic systems. There are several reasons for using a
stable distribution to describe a system. The first is that
there are solid theoretical reasons for expecting a non-Gaussian stable
model, e.g. reflection off a rotating mirror yielding a Cauchy distribution,
hitting times for a Brownian motion yielding a Levy distribution, the
gravitational field of stars yielding the Holtsmark distribution; see Feller (1971) and Uchaikin and Zolotarev (1999) for these and other examples. The
second reason is the Generalized Central Limit Theorem which states that
the only possible non-trivial limit of normalized sums of independent
identically distributed terms is stable. It is argued that
some observed quantities are the sum of many small terms - the price of
a stock, the noise in a communication system, etc. and hence a stable model should be used to describe such systems. The
third argument for modeling with stable distributions is empirical: many
large data sets exhibit heavy tails and skewness. The
strong empirical evidence for these features combined with the Generalized
Central Limit Theorem is used by many to justify the use of stable models. Examples
in finance and economics are given in Mandelbrot (1963), Fama (1965), Samuelson (1967) Roll (1970), Embrechts et al. (1997), Rachev and Mittnik (2000), McCulloch (1996); in communication
systems by Stuck and Kleiner (1974), Zolotarev (1986), and Nikias and Shao (1995). Such data sets, which are poorly described
by a Gaussian model, can be well described by a stable distribution.
