Modifications of the Rayleigh and Bingham Tests for Uniformity of Directions

The general machinery of Cordeiro and Ferrari (1991, Biometrika78, 573–582) and Chandra and Mukerjee (1991, J. Multivariate Anal.36, 103–112) provides modifications of score test statistics which bring the null distributions close to their large-sample asymptotic distributions. These modifications are calculated here for the Rayleigh and Bingham tests of uniformity on spheres of arbitrary dimension, Stiefel manifolds, rotation groups, Grassmann manifolds and complex projective spaces.