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=3D1 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 Special Manifolds and Related Multivariate Topics * Distributions on the Special Manifolds * Decompositions of the Special Manifolds * Distributional Problems in the Decomposition Theorems and the Sampling Theory * The Inference on the Parameters of the Matrix Langevin Distributions * Large Sample Asymptotic Theorems in Connection with Tests for Uniformity * Asymptotic Theorems for Concentrated Matrix Langevin Distributions * High Dimensional Asymptotic Theorems * Procrustes Analysis on the Special Manifolds * Density Estimation on the Special Manifolds * Measures of Orthogonal Association on the Special Manifolds
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