Symmetric Distributions for Dependent Unit Vectors

Abstract

This paper introduces several notions of symmetry for the joint distribution of two dependent unit vectors. Bivariate generalizations of $\mathscr{L}$-symmetry (Rivest, 1984) and rotational symmetry are introduced. If the joint distribution of two unit vectors is at least $\mathscr{L}$-symmetric the information matrix for the parameters indexing it is shown to have a simple shape.