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.

This book is concerned with statistical analysis on special manifoldsan area in multivariate statistics.

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.