Expanding the Domains and Methods for Creating Directional Cauchy Distributions

• Activity Number: 413
• Type: Contributed
• Date/Time: Tuesday, August 3, 2010 : 2:00 PM to 3:50 PM
• Sponsor: Section for Statistical Programmers and Analysts
• Author(s): Tom Downs*+
• Companies: Research Statistics, Inc.
• Keywords: Straight line projection ; Stereographic projection ; Spherical and hyperspherical Cauchy distributions

Abstract

The only domain for directional Cauchy distributions has been the unit circle. This family of Cauchy distributions is closed under changes in location and scale by rotations and straight line projections via circle mappings. Members of this family have been created using projections from extended lines or unit circles to unit circles, or by wrapping the linear Cauchy distribution about a unit circle. The domains for directional Cauchy distributions are expanded to include unit spheres and hyper-spheres of arbitrary dimension. All these new families are closed under changes in location and scale. Changes in location are done with rotations. Changes in scale are done with straight line projections from unit hyper-spheres to themselves, and by rescaling hyper-spheres via generalized stereographic projection.