Author | Chikuse, Yasuko. author |
---|---|
Title | Statistics on Special Manifolds [electronic resource] / by Yasuko Chikuse |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2003 |
Connect to | http://dx.doi.org/10.1007/978-0-387-21540-2 |
Descript | XXVI, 403 p. online resource |
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 the Parameter 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 Stamโs 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. Invariant Polynomials 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