Stable Distribution - Parameterizations

Parameterizations

The above definition is only one of the parameterizations in use for stable distributions; it is the most common but is not continuous in the parameters. For example, for the case α = 1 we could replace Φ by: (Nolan 2009)

and μ by

This parameterization has the advantage that we may define a standard distribution using

and

The pdf for all α will then have the following standardization property: