This book is concerned with statistical analysis on special manifolds, an area in multivariate statistics.
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Inhaltsangabe
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.
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.
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