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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.

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The asymptotic distribution of the Tukey median has recently been obtained by Nolan in a bivariate setting and by Bai and He in the general multivariate case. To establish their theorem, these authors made a strong symmetry hypothesis on the distribution and, in the case of Bai and He, assumed the existence of a density function with a gradient satisfying a finite-moment condition. This paper extends the above result by deriving the asymptotic distribution of the above location estimate without making any symmetry or differentiability assumption on the distribution. © 2001 Elsevier Science (USA) AMS 1991 subject classifications: 62F20; 60F05, 62G20.