學術產出-Theses
Article View/Open
Publication Export
-
題名 多變量d轉換的一些應用
Some applications of multivariate d-transformations作者 郭錕霖 貢獻者 姜志銘
郭錕霖關鍵詞 多變量d轉換
多變量d特徵函數
Dirichlet分配
對稱分配
極限分配
反演過程
multivariate d-transformation
multivariate d-characteristic function
Dirichlet distribution
spherical distribution
limiting distribution
inversion process
Carlson R日期 2002 上傳時間 9-May-2016 16:39:17 (UTC+8) 摘要 Jiang (1997) 首先提出多變量d轉換與其性質。利用多變量d轉換，我們可以定義新式的特徵函數，並且稱它們是多變量d特徵函數。在這篇論文中，我們將使用多變量d特徵函數來證明在普通的條件下，Dirichlet隨機向量的線性組合會分配收斂（converge in distribution）到一個對稱的分配。此外，當給定一個分配函數的多變量d特徵函數，我們將建構一個方法來決定此分配函數。另一方面，我們將證明多變量d特徵函數擁有很多類似傳統的特徵函數的性質。
A multivariate d-transformation and its properties were first given by Jiang (1997). By means of the multivariate d-transformations, we can define new kinds of characteristic functions and call them multivariate d-characteristic functions. In this thesis, we will use the multivariate d-characteristic function to show that the linear combinations of Dirichlet random vectors, under regularity conditions, converge in distribution to a spherical distribution. Moreover, We will construct a method for constructing the distribution function with a given multivariate d-characteristic function. In addition, we will show that the multivariate d-characteristic function has many properties which are similar to those of the traditional characteristic function.
書名頁 謝辭 Contents Abstract i 中文摘要 ii 1. Introduction 1 1.1 The c-characteristic function 1 1.2 The Ferguson-Dirichlet process 2 1.3 Compatible conditional distributions 4 1.4 Inverse Bayes formula 6 2. Notations and useful equations 7 2.1 Notations 7 2.2 Useful equations 8 3. The c-characteristic function 12 3.1 The univariate c-characteristic function 12 3.2 The multivariate c-characteristic function 13 3.3 Inversion formulas of a univariate c-characteristic function 16 3.4 Density construction through Fourier transformation 24 3.5 Distributions of linear combinations of the components of a Dirichlet random vector 25 4. The Ferguson-Dirichlet process 31 4.1 Random functionals of a Ferguson-Dirichlet process 31 4.2 The Ferguson-Dirichlet process over n-dimensional sphere 41 4.3 Proof of Theorem 4.5 48 4.4 Proof of Theorem 4.6 55 5. Compatible conditional distributions 59 5.1 The bivariate discrete case 59 5.2 The trivariate discrete case 77 5.3 The general discrete case 91 5.4 The bivariate continuous case 93 5.5 The general continuous case 98 6. Generalized inverse Bayes formula 102 6.1 The finite and discrete case 102 6.2 The continuous case 108 6.3 Algorithm 115 7. Conclusions 119 References 121參考文獻 Apostol, T. M. (1974), Mathematical Analysis, 2nd ed., Addison-Wesley. Carlson, B. C. (1977), Special Functions of Applied Mathematics, Academic Press, New York. Chung, Kai Lai (1968), A Course in Probability Theory, Harcourt,Brace and World, New York. Dickey, J. M. and Jiang, T. J. (1998), "Filtered-Variate Prior Distributions for Histogram Smoothing," Journal of the American Statistical Association, 93, pp. 651--662. Erdelyi, A. (1953) (Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.), Higher Transcendental Functions, vol. I, McGraw-Hill, New York. Friedberg, S. H., Insel, A. J., and Spence, L. E. (1992), Linear Algebra, 2nd ed., Prentice-Hall, New York. Gradshteyn, I. S. and Ryzbik, I. M.; Alan Jeffrey, editor; Translated from the Russian by Scripta Technica, Inc. (2000), Table of Integrals, Series, and Products, 6th ed., Academic Press, Boston. Hogg, Robert V. and Craig, Allen T. (1995), Introduction to Mathematical Statistics, 5th ed., Prentice-Hall, Englewood Cliffs, New Jersey. Jiang, J. (1988), "Starlike functions and linear functions of a Dirichlet distributed vector," SIAM J. Math. Anal., 19, pp. 390--397. Jiang, T. (1991), "Distribution of random functional of a Dirichlet process on the unit disk," Statistics and Probability Letters, 12, pp. 263--265. Jiang, J. (1997), "Multivariate d-transformations with applications," Taiwan National Science Council final report (NSC 85--2121--M--004--007). Jiang, T. (2002), "A new multivariate transformation and distribution of random functional of a Dirichlet process on the solid bounded by an ellipse." To be published. Lord, R. D. (1954), "The use of the Hankel transformations in statistics. I. General theory and examples," Biometrika, 41, pp. 44--55. Marsden, Jerrold E. and Hoffman, Michael J. (1993),Elementary Classical Analysis, 2nd ed., W. H. Freeman and Company, New York. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I. (1986), Integrals and Series, vol. 3, Gordon and Breach Science Publishers, New York. Roussas, George G. (1997), A Course in Mathematical Statistics, 2nd ed., Academic Press, San Diego, Calif. 描述 碩士
國立政治大學
應用數學系
88751006資料來源 http://thesis.lib.nccu.edu.tw/record/#A2010000222 資料類型 thesis dc.contributor.advisor 姜志銘 zh_TW dc.contributor.author (Authors) 郭錕霖 zh_TW dc.creator (作者) 郭錕霖 zh_TW dc.date (日期) 2002 en_US dc.date.accessioned 9-May-2016 16:39:17 (UTC+8) - dc.date.available 9-May-2016 16:39:17 (UTC+8) - dc.date.issued (上傳時間) 9-May-2016 16:39:17 (UTC+8) - dc.identifier (Other Identifiers) A2010000222 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/95619 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學系 zh_TW dc.description (描述) 88751006 zh_TW dc.description.abstract (摘要) Jiang (1997) 首先提出多變量d轉換與其性質。利用多變量d轉換，我們可以定義新式的特徵函數，並且稱它們是多變量d特徵函數。在這篇論文中，我們將使用多變量d特徵函數來證明在普通的條件下，Dirichlet隨機向量的線性組合會分配收斂（converge in distribution）到一個對稱的分配。此外，當給定一個分配函數的多變量d特徵函數，我們將建構一個方法來決定此分配函數。另一方面，我們將證明多變量d特徵函數擁有很多類似傳統的特徵函數的性質。 zh_TW dc.description.abstract (摘要) A multivariate d-transformation and its properties were first given by Jiang (1997). By means of the multivariate d-transformations, we can define new kinds of characteristic functions and call them multivariate d-characteristic functions. In this thesis, we will use the multivariate d-characteristic function to show that the linear combinations of Dirichlet random vectors, under regularity conditions, converge in distribution to a spherical distribution. Moreover, We will construct a method for constructing the distribution function with a given multivariate d-characteristic function. In addition, we will show that the multivariate d-characteristic function has many properties which are similar to those of the traditional characteristic function. en_US dc.description.abstract (摘要) 書名頁 謝辭 Contents Abstract i 中文摘要 ii 1. Introduction 1 1.1 The c-characteristic function 1 1.2 The Ferguson-Dirichlet process 2 1.3 Compatible conditional distributions 4 1.4 Inverse Bayes formula 6 2. Notations and useful equations 7 2.1 Notations 7 2.2 Useful equations 8 3. The c-characteristic function 12 3.1 The univariate c-characteristic function 12 3.2 The multivariate c-characteristic function 13 3.3 Inversion formulas of a univariate c-characteristic function 16 3.4 Density construction through Fourier transformation 24 3.5 Distributions of linear combinations of the components of a Dirichlet random vector 25 4. The Ferguson-Dirichlet process 31 4.1 Random functionals of a Ferguson-Dirichlet process 31 4.2 The Ferguson-Dirichlet process over n-dimensional sphere 41 4.3 Proof of Theorem 4.5 48 4.4 Proof of Theorem 4.6 55 5. Compatible conditional distributions 59 5.1 The bivariate discrete case 59 5.2 The trivariate discrete case 77 5.3 The general discrete case 91 5.4 The bivariate continuous case 93 5.5 The general continuous case 98 6. Generalized inverse Bayes formula 102 6.1 The finite and discrete case 102 6.2 The continuous case 108 6.3 Algorithm 115 7. Conclusions 119 References 121 - dc.description.tableofcontents 謝辭 Abstract-----i 中文摘要-----ii Contents 1 Introduction-----1 2 Multivariate d-transformation-----2 2.1 Univariate d-transformation-----2 2.2 Multivariate d-transformation-----6 2.3 Limiting distributions-----8 3 Elementary theorems-----17 4 Inversion formula-----31 4.1 Moments and moment generating function-----31 4.2 Traditional characteristic function-----33 4.3 Inversion process-----39 5 Some applications-----43 6 Conclusion-----49 References-----50 Appendices-----52 A To prove the identity of equation (2.7)-----52 B To prove the second identity of expression (4.18)-----53 C To prove the third identity of expression (4.18)-----54 D Univariate d-characteristic functions of some distributions in Section 3-----55 E To determine the probability density function of equation (5.4)-----58 zh_TW dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#A2010000222 en_US dc.subject (關鍵詞) 多變量d轉換 zh_TW dc.subject (關鍵詞) 多變量d特徵函數 zh_TW dc.subject (關鍵詞) Dirichlet分配 zh_TW dc.subject (關鍵詞) 對稱分配 zh_TW dc.subject (關鍵詞) 極限分配 zh_TW dc.subject (關鍵詞) 反演過程 zh_TW dc.subject (關鍵詞) multivariate d-transformation en_US dc.subject (關鍵詞) multivariate d-characteristic function en_US dc.subject (關鍵詞) Dirichlet distribution en_US dc.subject (關鍵詞) spherical distribution en_US dc.subject (關鍵詞) limiting distribution en_US dc.subject (關鍵詞) inversion process en_US dc.subject (關鍵詞) Carlson R en_US dc.title (題名) 多變量d轉換的一些應用 zh_TW dc.title (題名) Some applications of multivariate d-transformations en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) Apostol, T. M. (1974), Mathematical Analysis, 2nd ed., Addison-Wesley. Carlson, B. C. (1977), Special Functions of Applied Mathematics, Academic Press, New York. Chung, Kai Lai (1968), A Course in Probability Theory, Harcourt,Brace and World, New York. Dickey, J. M. and Jiang, T. J. (1998), "Filtered-Variate Prior Distributions for Histogram Smoothing," Journal of the American Statistical Association, 93, pp. 651--662. Erdelyi, A. (1953) (Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.), Higher Transcendental Functions, vol. I, McGraw-Hill, New York. Friedberg, S. H., Insel, A. J., and Spence, L. E. (1992), Linear Algebra, 2nd ed., Prentice-Hall, New York. Gradshteyn, I. S. and Ryzbik, I. M.; Alan Jeffrey, editor; Translated from the Russian by Scripta Technica, Inc. (2000), Table of Integrals, Series, and Products, 6th ed., Academic Press, Boston. Hogg, Robert V. and Craig, Allen T. (1995), Introduction to Mathematical Statistics, 5th ed., Prentice-Hall, Englewood Cliffs, New Jersey. Jiang, J. (1988), "Starlike functions and linear functions of a Dirichlet distributed vector," SIAM J. Math. Anal., 19, pp. 390--397. Jiang, T. (1991), "Distribution of random functional of a Dirichlet process on the unit disk," Statistics and Probability Letters, 12, pp. 263--265. Jiang, J. (1997), "Multivariate d-transformations with applications," Taiwan National Science Council final report (NSC 85--2121--M--004--007). Jiang, T. (2002), "A new multivariate transformation and distribution of random functional of a Dirichlet process on the solid bounded by an ellipse." To be published. Lord, R. D. (1954), "The use of the Hankel transformations in statistics. I. General theory and examples," Biometrika, 41, pp. 44--55. Marsden, Jerrold E. and Hoffman, Michael J. (1993),Elementary Classical Analysis, 2nd ed., W. H. Freeman and Company, New York. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I. (1986), Integrals and Series, vol. 3, Gordon and Breach Science Publishers, New York. Roussas, George G. (1997), A Course in Mathematical Statistics, 2nd ed., Academic Press, San Diego, Calif. zh_TW
