A Test Procedure for Uniformity on the Stiefel Manifold Based on Projection Toshiya Iwashita a, ∗ Bernhard Klar b , Moe Amagai c , Hiroki Hashiguchi d a Department of Liberal Arts, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki Noda, Chiba 278–8510, JAPAN b Institut f¨ ur Stochastik, Fakult¨ at f¨ ur Mathematik, Karlsruher Institut f¨ ur Technologie, Englerstraße 2, 76131 Karlsruhe, GERMANY c Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162–8601, JAPAN d Department of Mathematical Information Science, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162–8601, JAPAN ABSTRACT : This paper proposes a new procedure to test uniformity on the Stiefel manifold. As well as the theoretical analysis of the test procedure, numerical experiments are conducted to illustrate usage and efficiencies through the power under alternative hypotheses. Keywords: Anderson-Darling test; Goodness of fit test; Kolmogorov-Smirnov test; Spherical distribution; Stiefel manifold; Uniform distribution MSC: primary 62H11, secondary 62H15 1. Introduction Let X be a p × r (p ≥ r) random matrix which satisfies X ′ X = I r , where A ′ denotes the transpose of the matrix A, and I d is the d × d identity matrix. Consider the testing problem of the null hypothesis H 0 : X are uniformly distributed over V r (R p ), (1) where V r (R p ) stands for the Stiefel manifold of orthonormal r-frames in the Euclidean space R p . To address the hypothesis (1), Jupp (2001) reviewed the Rayleigh test and proposed a procedure based on the modified Rayleigh’s statistic defined by S ∗ = ( 1 - 1 2N + 1 2(pr + 2)N S ) S, (2) * Corespondent author. E-mail : iwashita@rs.noda.tus.ac.jp (T. Iwashita) 1