ABSTRACT
Probability Density Function (PDF) closed expression of stable distribution does not exist, other than Gaussian distribution, Cauchy distribution and Levy distribution, so stable distribution is usually characterized by characteristic function. Mathematicians were interested in parameterizations that the characteristic function has the simplest form, namely, standard parameterization. However, standard parameterization used to stable distribution is not suitable for numerical calculation and modeling. In addition, there are multiple parameterizations and various parameterizations make researchers confused. For example, Chambers et al. (1976) study the stable random number generator, which has a slightly incorrect form: Most researchers expect to get a S (1, ,0)βα result, however, the routine actually returns random variables with a S (1, , tan 2)β β piα−α distribution when 1α ≠ , and a S (1, ,0)βα distribution when 1α = . And another example is that Hall (1981) describes a “comedy of errors” caused by parameterization choices. The most common mistake concerns the sign of the skewness parameter when
= 1 (Holt & Crow 1973).
