A statistical model for random rotations

This paper studies the properties of the Cayley distributions, a new family of models for random p × p rotations. This class of distributions is related to the Cayley transform that maps a p ( p - 1 ) / 2 × 1 vector s into SO ( p ) , the space of p × p rotation matrices. First an expression for the uniform measure on SO ( p ) is derived using the Cayley transform, then the Cayley density for random rotations is investigated. A closed-form expression is derived for its normalizing constant, a simple simulation algorithm is proposed, and moments are derived. The efficiencies of moment estimators of the parameters of the new model are also calculated. A Monte Carlo investigation of tests and of confidence regions for the parameters of the new density is briefly summarized. A numerical example is presented.