特殊多様体上の統計解析：特に分布論の考察
この論文では，著者の英語で書かれた本Chikuse (2003)で扱われた特殊多様体，特にスティーフェル多様体とグラスマン多様体上の統計解析を紹介する．特に，多様体上の分布論に関する解析を論じる：これら多様体の定義と関連する結果，母集団統計分布，種々の分解定理とそれに伴う不変測度の分解と分布，そして漸近理論など．これら解析結果の詳細や証明，多様体上の他の解析法，そして，結果の導出に有用な行列変数の不変多項式論については原本を参照してほしい．...
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| Published in | 日本統計学会誌 Vol. 48; no. 1; pp. 71 - 88 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | Japanese |
| Published |
一般社団法人 日本統計学会
26.09.2018
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0389-5602 2189-1478 |
| DOI | 10.11329/jjssj.48.71 |
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| Abstract | この論文では，著者の英語で書かれた本Chikuse (2003)で扱われた特殊多様体，特にスティーフェル多様体とグラスマン多様体上の統計解析を紹介する．特に，多様体上の分布論に関する解析を論じる：これら多様体の定義と関連する結果，母集団統計分布，種々の分解定理とそれに伴う不変測度の分解と分布，そして漸近理論など．これら解析結果の詳細や証明，多様体上の他の解析法，そして，結果の導出に有用な行列変数の不変多項式論については原本を参照してほしい． |
|---|---|
| AbstractList | この論文では，著者の英語で書かれた本Chikuse (2003)で扱われた特殊多様体，特にスティーフェル多様体とグラスマン多様体上の統計解析を紹介する．特に，多様体上の分布論に関する解析を論じる：これら多様体の定義と関連する結果，母集団統計分布，種々の分解定理とそれに伴う不変測度の分解と分布，そして漸近理論など．これら解析結果の詳細や証明，多様体上の他の解析法，そして，結果の導出に有用な行列変数の不変多項式論については原本を参照してほしい． |
| Author | 筑瀬, 靖子 |
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| Copyright | 2018 日本統計学会 |
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| DOI | 10.11329/jjssj.48.71 |
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| References | Fang, K. T. and Zhang, Y. T. (1990). Generalized Multivariate Analysis, Science Press Beijing and Springer-Verlag, Berlin. Mardia, K. V. (1972). Statistics of Directional Data, Academic Press, New York. Downs, T. D. (1972). Orientation statistics, Biometrika, 59, 665–676. Watson, G. S. and Williams, E. J. (1956). On the construction of significance tests on the circle and the sphere, Biometrika, 43, 344–352. Mardia, K. V. and Holmes, D. (1980). A statistical analysis of megalithic data under elliptic pattern, J. R. Statist. Soc. A, 143, 293–302. Stam, A. J. (1982). Limit theorems for uniform distributions on spheres in high dimensional Euclidean spaces, J. Appl. Prob., 19, 221–228. Prentice, M. J. (1982). Antipodally symmetric distributions for orientation statistics, J. Statist. Plan. Inf., 6, 205–214. Chikuse, Y. and Watson, G. S. (1995). Large sample asymptotic theory of tests for uniformity on the Grassmann manifold, J. Multivariate Anal., 54, 18–31. Jupp, P. E. and Mardia, K. V. (1979). Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions, Ann. Statist., 7, 599–606. Fisher, R. A. (1953). Dispersion on a sphere, Proc. R. Soc. Lond. A, 217, 295–305. Langevin, P. (1905). Magnétisme et théorie des électrons, Ann. de Chim. et de Phys., 5, 70–127. Chikuse, Y. (1994b). Asymptotic theorems and tests for uniformity on the high dimensional Stiefel and Grassmann manifolds, Research Paper, No. 49, Kagawa University. Khatri, C. G. and Mardia, K. V. (1977). The von Mises-Fisher matrix distribution in orientation statistics, J. R. Statist. Soc. B, 39, 95–106. Bingham, C. (1974). An antipodally symmetric distribution on the sphere, Ann. Statist., 2, 1201–1225. Chikuse, Y. (1994a). Invariant measures on Stiefel manifolds with applications to multivariate analysis, in Multivariate Analysis and Its Applications, (T. W. Anderson, K. T. Fang, and I. Olkin, eds.), IMS Lectures Notes Monograph Series, 24, pp. 177–193. Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics, John Wiley & Sons, New York. Tyler, D. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere, Biometrika, 74, 579–589. Chikuse, Y. (1991a). High dimensional limit theorems and matrix decompositions on the Stiefel manifold, J. Multivariate Anal., 36, 145–162. Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1987). Statistical Analysis of Spherical Data, Cambridge University Press, London and New York. von Mises, R. (1918). Über die ``Ganzzahligkeit'' der Atomgewicht und verwandte Fragen, Physikalische Z., 19, 490–500. Watson, G. S. (1983). Statistics on Spheres, University of Arkansas Lecture Notes in the Mathematical Sciences, 6, John Wiley & Sons, New York. Chikuse, Y. (1990). Distributions of orientations on Stiefel manifolds, J. Multivariate Anal., 33, 247–264. Chikuse, Y. (1991b). Asymptotic expansions for distributions of the large sample matrix resultant and related statistics on the Stiefel manifold, J. Multivariate Anal., 39, 270–283. Chikuse, Y. (1993). High dimensional asymptotic expansions for the matrix Langevin distributions on the Stiefel manifold, J. Multivariate Anal., 44, 82–101. Anderson, D. A. (1981). The circular structural model, J. R. Statist. Soc. B, 43, 131–141. James, A. T. (1954). Normal multivariate analysis and the orthogonal group, Ann. Math. Statist., 25, 40–75. Arnold, K. J. (1941). On spherical probability distributions, Ph.D. Thesis, Massachusetts Institute of Technology. Chikuse, Y. (2003). Statistics on Special Manifolds, Lecture Notes in Statistics, 174, Springer, New York. |
| References_xml | – reference: Downs, T. D. (1972). Orientation statistics, Biometrika, 59, 665–676. – reference: Chikuse, Y. (2003). Statistics on Special Manifolds, Lecture Notes in Statistics, 174, Springer, New York. – reference: Mardia, K. V. and Holmes, D. (1980). A statistical analysis of megalithic data under elliptic pattern, J. R. Statist. Soc. A, 143, 293–302. – reference: von Mises, R. (1918). Über die ``Ganzzahligkeit'' der Atomgewicht und verwandte Fragen, Physikalische Z., 19, 490–500. – reference: Chikuse, Y. (1993). High dimensional asymptotic expansions for the matrix Langevin distributions on the Stiefel manifold, J. Multivariate Anal., 44, 82–101. – reference: Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics, John Wiley & Sons, New York. – reference: James, A. T. (1954). Normal multivariate analysis and the orthogonal group, Ann. Math. Statist., 25, 40–75. – reference: Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1987). Statistical Analysis of Spherical Data, Cambridge University Press, London and New York. – reference: Stam, A. J. (1982). Limit theorems for uniform distributions on spheres in high dimensional Euclidean spaces, J. Appl. Prob., 19, 221–228. – reference: Fisher, R. A. (1953). Dispersion on a sphere, Proc. R. Soc. Lond. A, 217, 295–305. – reference: Watson, G. S. (1983). Statistics on Spheres, University of Arkansas Lecture Notes in the Mathematical Sciences, 6, John Wiley & Sons, New York. – reference: Mardia, K. V. (1972). Statistics of Directional Data, Academic Press, New York. – reference: Anderson, D. A. (1981). The circular structural model, J. R. Statist. Soc. B, 43, 131–141. – reference: Jupp, P. E. and Mardia, K. V. (1979). Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions, Ann. Statist., 7, 599–606. – reference: Watson, G. S. and Williams, E. J. (1956). On the construction of significance tests on the circle and the sphere, Biometrika, 43, 344–352. – reference: Khatri, C. G. and Mardia, K. V. (1977). The von Mises-Fisher matrix distribution in orientation statistics, J. R. Statist. Soc. B, 39, 95–106. – reference: Prentice, M. J. (1982). Antipodally symmetric distributions for orientation statistics, J. Statist. Plan. Inf., 6, 205–214. – reference: Langevin, P. (1905). Magnétisme et théorie des électrons, Ann. de Chim. et de Phys., 5, 70–127. – reference: Bingham, C. (1974). An antipodally symmetric distribution on the sphere, Ann. Statist., 2, 1201–1225. – reference: Fang, K. T. and Zhang, Y. T. (1990). Generalized Multivariate Analysis, Science Press Beijing and Springer-Verlag, Berlin. – reference: Chikuse, Y. (1994b). Asymptotic theorems and tests for uniformity on the high dimensional Stiefel and Grassmann manifolds, Research Paper, No. 49, Kagawa University. – reference: Chikuse, Y. (1990). Distributions of orientations on Stiefel manifolds, J. Multivariate Anal., 33, 247–264. – reference: Chikuse, Y. (1991b). Asymptotic expansions for distributions of the large sample matrix resultant and related statistics on the Stiefel manifold, J. Multivariate Anal., 39, 270–283. – reference: Chikuse, Y. and Watson, G. S. (1995). Large sample asymptotic theory of tests for uniformity on the Grassmann manifold, J. Multivariate Anal., 54, 18–31. – reference: Arnold, K. J. (1941). On spherical probability distributions, Ph.D. Thesis, Massachusetts Institute of Technology. – reference: Chikuse, Y. (1994a). Invariant measures on Stiefel manifolds with applications to multivariate analysis, in Multivariate Analysis and Its Applications, (T. W. Anderson, K. T. Fang, and I. Olkin, eds.), IMS Lectures Notes Monograph Series, 24, pp. 177–193. – reference: Tyler, D. (1987). Statistical analysis for the angular central Gaussian distribution on the sphere, Biometrika, 74, 579–589. – reference: Chikuse, Y. (1991a). High dimensional limit theorems and matrix decompositions on the Stiefel manifold, J. Multivariate Anal., 36, 145–162. |
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| SubjectTerms | Bingham分布 Langevin分布 Orientation グラスマン多様体 スティーフェル多様体 一様分布 漸近理論 |
| Title | 特殊多様体上の統計解析：特に分布論の考察 |
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