This book is concerned with statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, treated as statistical sample spaces consisting of matrices. The former is represented by the set of m x k matrices whose columns are mutually orthogonal k-variate vectors of unit length, and the latter by the set of m x m orthogonal projection matrices idempotent of rank k. The observations for the special case k=1 are regarded as directed vectors on a unit hypersphere and as axes or lines undirected, respectively. Statistical analysis on these manifolds is required, especially for low dimensions in practical applications, in the earth (or geological) sciences, astronomy, medicine, biology, meteorology, animal behavior and many other fields. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on the manifold must make some contributions to the related sciences. The reader may already know the usual theory of multivariate analysis on the real Euclidean space and intend to deeper or broaden the research area to statistics on special manifolds, which is not treated in general textbooks of multivariate analysis.

1. The Special Manifolds and Related Multivariate Topics.- 1.1. Introduction.- 1.2. Analytic Manifolds and Related Topics.- 1.3. The Special Stiefel and Grassmann Manifolds.- 1.4. The Invariant Measures on the Special Manifolds.- 1.5. Jacobians and Some Related Multivariate Distributions.- 2. Distributions on the Special Manifolds.- 2.1. Introduction.- 2.2. Properties of the Uniform Distributions.- 2.3. Non-uniform Distributions.- 2.4 Random Distributions of the Orientations of a Matrix.- 2.5. Simulation Methods for Generating Pseudo-Random Matrices on Vk,m and Pk,m?k.- 3. Decompositions of the Special Manifolds.- 3.1. Introduction.- 3.2. Decompositions onto Orthogonally Subspaces of Vk,m.- 3.3. Other Decompositions of Vk,m.- 3.4. One-to-One Transformations of Pk,m?k onto Rm?k,k or Rm?k,k(1).- 3.5. Another Decomposition of Pk,m?k (or Gk,m?k).- 4. Distributional Problems in the Decomposition Theorems and the Sampling Theory.- 4.1. Introduction.- 4.2. Distributions of the Component Matrix Variates in the Decompositions of the Special Manifolds.- 4.3. Distributions of Canonical Correlation Coefficients of General Dimension.- 4.4. General Families of Distributions on Vk,m and Pk,m?k.- 4.5. Sampling Theory for the Matrix Langevin Distributions.- 5. The Inference on the Parameters of the Matrix Langevin Distributions.- 5.1. Introduction.- 5.2. Fisher Scoring Methods on Vk,m.- 5.3. Other Topics in the Inference on the Orientation Parameters on Vk,m.- 5.4. Fisher Scoring Methods on Pk,m?k.- 5.5. Other Topics in the Inference on the Orientation Parameter on Pk,m?k.- 6. Large Sample Asymptotic Theorems in Connection with Tests for Uniformity.- 6.1. Introduction.- 6.2. Asymptotic Expansions for the Sample Mean Matrix on Vk,m.- 6.3. Asymptotic Properties of theParameter Estimation and the Tests for Uniformity on Vk,m.- 6.4. Asymptotic Expansions for the Sample Mean Matrix on Pk,m?k.- 6.5. Asymptotic Properties of the Parameter Estimation and the Tests for Uniformity on Pk,m?k.- 7. Asymptotic Theorems for Concentrated Matrix Langevin Distributions.- 7.1. Introduction.- 7.2. Estimation of Large Concentration Parameters.- 7.3. Asymptotic Distributions in Connection with Testing Hypotheses of the Orientation Parameters on Vk,m.- 7.4. Asymptotic Distributions in Connection with Testing Hypotheses of the Orientation Parameter on Pk,m?k.- 7.5. Classification of the Matrix Langevin Distributions.- 8. High Dimensional Asymptotic Theorems.- 8.1. Introduction.- 8.2. Asymptotic Expansions for the Matrix Langevin Distributions on Vk,m.- 8.3. Asymptotic Expansions for the Matrix Bingham and Langevin Distributions on Vk,m and Pk,m?k.- 8.4. Generalized Stam's Limit Theorems.- 8.5. Asymptotic Properties of the Parameter Estimation and the Tests of Hypotheses.- 9. Procrustes Analysis on the Special Manifolds.- 9.1. Introduction.- 9.2. Procrustes Representations of the Manifolds.- 9.3. Perturbation Theory.- 9.4. Embeddings.- 10. Density Estimation on the Special Manifolds.- 10.1. Introduction.- 10.2. Kernel Density Estimation on Pk,m?k.- 10.3. Kernel Density Estimation on Vk,m.- 10.4. Density Estimation via the Decompositions (or Transformations) of Pk,m?k and Vk,m.- 10.5. Density Estimation on the Spaces Sm and Rm,p.- 11. Measures of Orthogonal Association on the Special Manifolds.- 11.1. Introduction.- 11.2. Measures of Orthogonal Association on Vk,m.- 11.3. Measures of Orthogonal Association on Pk,m?k.- 11.4. Distributional and Sampling Problems on Vk,m.- 11.5. Related Regression Models on Vk,m.- Appendix A. InvariantPolynomials with Matrix Arguments.- A.1. Introduction.- A.2. Zonal Polynomials.- A.3. Invariant Polynomials with Multiple Matrix Arguments.- A.4. Basic Properties of Invariant Polynomials.- A.5. Special Cases of Invariant Polynomials.- A.6. Hypergeometric Functions with Matrix Arguments.- A.7. Tables of Zonal and Invariant Polynomials.- Appendix B. Generalized Hermite and Laguerre Polynomials with Matrix Arguments.- B.1. Introduction.- B.2.1. Series (Edgeworth) Expansions for Multiple Random Symmetric Matrices.- B.3.1. Series (Edgeworth) Expansions for Multiple Random Rectangular Matrices.- B.4. Generalized Laguerre Polynomials in Multiple Matrices.- B.4.1. Generalized (Central) Laguerre Polynomials.- B.4.2. Generalized Noncentral Laguerre Polynomials.- B.5. Generalized Multivariate Meixner Classes of Invariant Distributions of Multiple Random Matrices.- Appendix C. Edgeworth and Saddle-Point Expansions for Random Matrices.- C.1. Introduction.- C.2. The Case of Random Symmetric Matrices.- C.2.1. Edgeworth Expansions.- C.2.2. Saddle-Point Expansions.- C.2.3. Generalized Edgeworth Expansions.- C.3. The Case of Random Rectangular Matrices.- C.3.1. Edgeworth Expansions.- C.3.2. Saddle-Point Expansions.- C.3.3. Generalized Edgeworth Expansions.- C.4. Applications.- C.4.1. Exact Saddle-Point Approximations.