A statistical model for random rotations

This paper studies the properties of the Cayley distributions, a new family of models for random pxp rotations. This class of distributions is related to the Cayley transform that maps a p(p-1)/2x1 vector s into SO(p), the space of pxp 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.