Statistics on Special Manifolds

1. Front Matter

   Pages N2-xxvi

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2. The Special Manifolds and Related Multivariate Topics

   • Yasuko Chikuse

   Pages 1-26

3. Distributions on the Special Manifolds

   • Yasuko Chikuse

   Pages 27-51

4. Decompositions of the Special Manifolds

   • Yasuko Chikuse

   Pages 53-80

5. Distributional Problems in the Decomposition Theorems and the Sampling Theory

   • Yasuko Chikuse

   Pages 81-108

6. The Inference on the Parameters of the Matrix Langevin Distributions

   • Yasuko Chikuse

   Pages 109-132

7. Large Sample Asymptotic Theorems in Connection with Tests for Uniformity

   • Yasuko Chikuse

   Pages 133-163

8. Asymptotic Theorems for Concentrated Matrix Langevin Distributions

   • Yasuko Chikuse

   Pages 165-186

9. High Dimensional Asymptotic Theorems

   • Yasuko Chikuse

   Pages 187-230

10. Procrustes Analysis on the Special Manifolds

    • Yasuko Chikuse

    Pages 231-246

11. Density Estimation on the Special Manifolds

    • Yasuko Chikuse

    Pages 247-282

12. Measures of Orthogonal Association on the Special Manifolds

    • Yasuko Chikuse

    Pages 283-294

13. Back Matter

    Pages 295-403

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The special manifolds of interest in this book are the Stiefel manifold and the Grassmann manifold. Formally, the Stiefel manifold Vk,m is the space of k­ frames in the m-dimensional real Euclidean space Rm, represented by the set of m x k matrices X such that X' X = I , where Ik is the k x k identity matrix, k and the Grassmann manifold Gk,m-k is the space of k-planes (k-dimensional hyperplanes) in Rm. We see that the manifold Pk,m-k of m x m orthogonal projection matrices idempotent of rank k corresponds uniquely to Gk,m-k. This book is concerned with statistical analysis on the manifolds Vk,m and Pk,m-k as statistical sample spaces consisting of matrices. The discussion is carried out on the real spaces so that scalars, vectors, and matrices treated in this book are all real, unless explicitly stated otherwise. For the special case k = 1, the observations from V1,m and G1,m-l are regarded as directed vectors on a unit sphere and as undirected axes or lines, respectively. There exists a large literature of applications of directional statis­ tics and its statistical analysis, mostly occurring for m = 2 or 3 in practice, in the Earth (or Geological) Sciences, Astrophysics, Medicine, Biology, Meteo­ rology, Animal Behavior, and many other fields. Examples of observations on the general Grassmann manifold Gk,m-k arise in the signal processing of radar with m elements observing k targets.

• correlation
• manifold
• mathematical statistics
• statistical inference
• statistics

• Faculty of Engineering, Kagawa University, Takamatsu, Kagawa, Japan

  Yasuko Chikuse

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