Multivariate asymptotic test of pairwise independence for orientations with the same symmetry group

This work addresses the problem of testing independence in r-tuples of orientations of symmetric objects in three-dimensional space. Such objects naturally occur in crystallography, materials science, and other fields. We propose a novel definition of covariance between two random orientations and utilise it in the construction of multivariate asymptotic tests for pairwise uncorrelatedness based on U-statistics. We evaluate the finite-sample performance of these tests through a simulation study that investigates their power for three models of r-tuples of random orientations and compares them with permutation tests based on sample covariances and total distance multivariance. The application of the proposed tests is demonstrated on a dataset of polycrystalline material with cubic symmetry of the crystal lattice.