A New Unified Approach for the Simulation of a Wide Class of Directional Distributions

ABSTRACT

The need for effective simulation methods for directional distributions has grown as they have become components in more sophisticated statistical models. A new acceptance–rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown that the proposed method has high efficiency and is also straightforward to use. Next, the simulation method is extended to the Fisher and Fisher–Bingham distributions on spheres and related manifolds. Together, these results provide a widely applicable and efficient methodology to simulate many of the standard models in directional data analysis. An R package simdd, available in the online supplementary material, implements these simulation methods.

KEYWORDS:

• Acceptance–rejection
• Angular central Gaussian distribution
• Bingham distribution
• Bivariate von Mises sine distribution
• Matrix Fisher distribution
• Simulation efficiency

Supplementary Materials

• R-package simdd: Code to implement the methods described in the article for the simulation of directional distributions (GNU zipped tar file).

Acknowledgments

The work of the first author for the FB5e distribution has been supported by the Air Force Office of Scientific Research, Air Force Material Command, USAF under Award No. FA9550-16-1-0099.