學術產出-學位論文

題名 研究Ferguson-Dirichlet過程和條件分配族相容性之新工具
New tools for studying the Ferguson-Dirichlet process and compatibility of a family of conditionals
作者 郭錕霖
Kuo,Kun Lin
貢獻者 姜志銘
Jiang,Thomas J.
郭錕霖
Kuo,Kun Lin
關鍵詞 c-特徵函數
相容性
Ferguson-Dirichlet過程
廣義貝氏反演公式
隨機泛函
c-characteristic function
compatibility
Ferguson-Dirichlet process
generalized inverse Bayes formula
random functional
日期 2007
上傳時間 17-九月-2009 13:50:18 (UTC+8)
摘要 單變量c-特徵函數已被證明可處理一些難以使用傳統特徵函數解決的問題,
在本文中,我們首先提出其反演公式,透過此反演公式,我們獲得(1)Dirichlet隨機向量之線性組合的機率密度函數;(2)以一些有趣測度為參數之Ferguson-Dirichlet過程其隨機動差的機率密度函數;(3)Ferguson-Dirichlet過程之隨機泛函的Lebesgue積分表示式。

本文給予對稱分配之多變量c-特徵函數的新性質,透過這些性質,我們證明在任何$n$維球面上之Ferguson-Dirichlet過程其隨機均值是一對稱分配,並且我們亦獲得其確切的機率密度函數,此外,我們將這些結果推廣至任何n維橢球面上。

我們亦探討條件分配相容性的問題,這個問題在機率理論與貝式計算上有其重要性,我們提出其充要條件。當給定相容的條件分配時,我們不但解決相關聯合分配唯一性的問題,而且也提供方法去獲得所有可能的相關聯合分配,我們亦給予檢驗相容性、唯一性及建構機率密度函數的演算法。

透過相容性的相關理論,我們提出完整且清楚地統合性貝氏反演公式理論,並建構可應用於一般測度空間的廣義貝氏反演公式。此外,我們使用廣義貝氏反演公式提供一個配適機率密度函數的演算法,此演算法沒有疊代演算法(如Gibbs取樣法)的收斂問題。
The univariate c-characteristic function has been shown to be important in cases that are hard to manage using the traditional characteristic function. In this thesis, we first give its inversion formulas. We then use them to obtain (1) the probability density functions (PDFs) of a linear combination of the components of a Dirichlet random vector; (2) the PDFs of random functionals of a Ferguson-Dirichlet process with some interesting parameter measures; (3) a Lebesgue integral expression of any random functional
of the Ferguson-Dirichlet process.

New properties of the multivariate c-characteristic function with a spherical distribution are given in this thesis. With them, we show that the random mean of a Ferguson-Dirichlet process over a spherical surface in n dimensions has a spherical distribution on the n-dimensional ball. Moreover, we derive its exact PDF. Furthermore, we generalize this result to any ellipsoidal surface in n-space.

We also study the issue of compatibility for specified conditional distributions. This issue is important in probability theory and Bayesian computations. Several necessary and sufficient conditions for the compatibility are provided. We also address the problem of uniqueness of the associated joint distribution when the given conditionals are compatible. In addition, we provide a method to obtain all possible joint distributions that have the given compatible conditionals. Algorithms for checking the compatibility and the uniqueness, and for constructing all associated densities are also given.

Through the related compatibility theorems, we provide a fully and cleanly unified theory of inverse Bayes formula (IBF) and construct a generalized IBF (GIBF) that is applicable in the more general measurable space. In addition, using the GIBF, we provide a marginal density fitting algorithm, which avoids the problems of convergence in iterative algorithm such as the Gibbs sampler.
參考文獻 Amemiya, T. (1975) Qualitative response models. Annals of Economic and Social Measurement, 4, 363-372.
Arnold, B.C. and Gokhale, D.V. (1998) Distributions of the most nearly compatible with given families of conditional distributions. Test, 7, 377-390.
Arnold, B.C. and Press, S.J. (1989) Compatible conditional distributions. J. Amer. Statist. Assoc., 84, 152-156.
Arnold, B.C., Castillo, E. and Sarabia, J.M. (2001) Conditionally specified distributions: an introduction
(with discussion). Statist. Sci., 16, 249-274.
Arnold, B.C., Castillo, E., and Sarabia, J.M. (2002) Exact and near compatibility of discrete conditional distributions. Comput. Statist. Data Anal., 40, 231-252.
Arnold, B.C., Castillo, E., and Sarabia, J.M. (2004) Compatibility of partial or complete conditional probability
specifications. J. Statist. Plann. Inference, 123, 133-159.
Carlson, B.C. (1977) Special Functions of Applied Mathematics. New York: Academic Press.
Casella, G. and George, E.I. (1992) Explaining the Gibbs sampler. Amer. Statist., 46, 167-174.
Chung, K.L. (1974) A Course in Probability Theory. New York: Academic Press.
Cifarelli, D.M. and Regazzini, E. (1990) Distribution functions of means of a Dirichlet process. Ann. Statist., 18, 429-442. Correction (1994): Ann. Statist., 22, 1633-1634.
Cifarelli, D.M. and Melilli, E. (2000) Some new results for Dirichlet Priors. Ann. Statist., 28, 1390-1413.
Dempster, A.P., Laird, N.M., and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B, 39, 1-38.
Diaconis, P. and Kemperman, J. (1996) Some new tools for Dirichlet priors. In J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith (eds.), Bayesian Statistics 5, pp. 97-106. Oxford University Press.
Dickey, J.M., Jiang, T.J., and Kuo, K.-L. (2008), Functionals of a Ferguson-Dirichlet process. Preprint.
Epifani, I., Guglielmi, A., and Melilli, E. (2006) A stochastic equation for the law of the random Dirichlet variance. Statist. Probab. Lett., 76 , 495-502.
Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1953) Higher Transcendental Functions, vol. I. New York: McGraw-Hill.
Ferguson, T.S. (1973) A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209-230.
Gelfand, A.E. and Smith, A.F.M. (1990) Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc., 85, 398-409.
Gourieroux, C. and Montfort, A. (1979) On the characterization of a joint probability distribution by conditional distributions. J. Econometrics, 10, 115-118.
Gradshteyn, I.S. and Ryzhik, I.M. (2000) Table of Integrals, Series, and Products, 6th ed. New York: Academic Press.
Grobner, W. and Hofreiter, W. (1973) Integraltafel, Vol. 2, 5th ed. New York: Springer-Verlag.
Hannum, R.C., Hollander, M., and Langberg, N.A. (1981) Distributional results for random functionals of a Dirichlet process. Ann. Probab., 9, 665-670.
Hjort, N.L. and Ongaro, A. (2005) Exact inference for random Dirichlet means. Stat. Inference Stoch. Process., 8, 227-254.
Jiang, J. (1988) Starlike functions and linear functions of a Dirichlet distributed vector. SIAM J. Math. Anal., 19, 390-397.
Jiang, T.J. (1991) Distribution of random functional of a Dirichlet process on the unit disk. Statist. Probab. Lett., 12, 263-265.
Jiang, T.J., Dickey, J.M., and Kuo, K.-L. (2004) A new multivariate transform and the distribution of a
random functional of a Ferguson-Dirichlet process.
Stochastic Process. Appl., 111, 77-95.
Jiang, T.J. and Kuo, K.-L. (2008a), Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere. To appear in Electron. Comm. Probab..
Jiang, T.J. and Kuo, K.-L. (2008b), The inversion formula of the c-characteristic function and its applications.
Preprint.
Kuo, K.-L. (2002) Some applications of multivariate c-transformations. Master thesis, Department of Mathematical Sciences, National Chengchi University.
Kuo, K.-L., Song, C.-C., and Jiang, T.J. (2008a),
Compatibility of discrete conditionals in higher dimensions. Preprint.
Kuo, K.-L., Song, C.-C., and Jiang, T.J. (2008b), Compatible continuous conditionals and an application on normal conditionals. Preprint.
Kuo, K.-L., Song, C.-C., and Jiang, T.J. (2008c), Generalized inverse Bayes formula for compatible conditional distributions. Preprint.
Lijoi, A. and Regazzini, E. (2004) Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab., 32, 1469-1495.
Liu, J.S. (1996) Discussion of "Statistical Inference and Monte Carlo Algorithms" by G. Casella. Test, 5, 305-310.
Lord, R.D. (1954) The use of the Hankel transformations in statistics. I. General theory and example. Biometrika, 41, 44-55.
Minc, H. (1988) Nonnegative Matrices. New York: Wiley.
Nerlove, M. and Press, S.J. (1986) Multivariate log-linear probability models in econometrics. In Advances in Statistical Analysis and Statistical Computing (Edited by Mariano, R. S.), 117-171. Greenwich, CT: JAI Press.
Ng, K.W. (1995) Explict formulas for unconditional pdf.
Research Report, No. 82 (revised). Department of Statistics, University of Hong Kong.
Ng, K.W. (1997) Inversion of Bayes formula: explict formulae
for unconditional pdf. In Advance in the Theory and Practice in Statistics (Edited by Johnson, N. L. and Balakrishnan, N.), 571-584, New York: Wiley.
Perez-Villalta, R. (2000) Variables finitas condicionalmente especificadas. Questioo, 24, 425-448.
Provost, S.B. and Cheong, Y.-H. (2000) On the distribution of linear combinations of the components of a Dirichlet random vector. Canad. J. Statist., 28, 417-425.
Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I. (1986)
Integrals and Series, Vol. 3. New York: Gordon and Breach Science Publishers.
Rao, C.R. (1973) Linear Statistical Inference and Its Applications, New York: Wiley.
Regazzini, E., Guglielmi, A., and Di Nunno, G. (2002)
Theory and numerical analysis for exact distributions of
functionals of a Dirichlet process. Ann. Statist., 30, 1376-1411.
Song, C.-C., Li, L.-A., Chen, C.-H., Jiang, T.J., and Kuo, K.-L. (2006), Compatibility of finite discrete conditional distributions. Under revision.
Sumner, D.B. (1949) An inversion formula for the generalized Stieltjes transform. Bull. Amer. Math. Soc., 55, 174-183.
Tan, M., Tian, G.-L. and Ng, K. W. (2003).A noniterative sampling method for computing posteriors in the structure of EM-type algorithms. Statist. Sinica, 13, 625-639.
Tanner, M.A. and Wong, W.H. (1987) The calculation of posterior distributions by data augmentation (with discussion). J. Amer. Statist. Assoc., 82, 528-540.
Tian, G.-L., Ng, K.W., and Geng, Z. (2003) Bayesian computation for contingency tables with incomplete cell-counts. Statist. Sinica, 13, 189-206.
Tian, G.-L. and Tan, M. (2003) Exact statistical solutions using the inversion Bayes formulae. Statist. Probab. Lett., 62, 305-315.
Tian, G.-L., Tan, M. and Ng, K.W. (2007) An exact non-iterative sampling procedure for discrete missing data problems. Statist. Neerlandica, 61, 232-242.
Weisstein, E.W. (2005) Permutation Matrix. From MathWorld-A Wolfram Web Resource.
http://mathworld.wolfram.com/PermutationMatrix.html
Widder, D.V. (1946) The Laplace Transform. Princenton University Press.
Yamato, H. (1984) Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab., 12, 262-267.
Zayed, A.I. (1996) Handbook of function and generalized function transformations. New York: CRC Press.
描述 博士
國立政治大學
應用數學研究所
91751501
96
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0917515011
資料類型 thesis
dc.contributor.advisor 姜志銘zh_TW
dc.contributor.advisor Jiang,Thomas J.en_US
dc.contributor.author (作者) 郭錕霖zh_TW
dc.contributor.author (作者) Kuo,Kun Linen_US
dc.creator (作者) 郭錕霖zh_TW
dc.creator (作者) Kuo,Kun Linen_US
dc.date (日期) 2007en_US
dc.date.accessioned 17-九月-2009 13:50:18 (UTC+8)-
dc.date.available 17-九月-2009 13:50:18 (UTC+8)-
dc.date.issued (上傳時間) 17-九月-2009 13:50:18 (UTC+8)-
dc.identifier (其他 識別碼) G0917515011en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/32607-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 91751501zh_TW
dc.description (描述) 96zh_TW
dc.description.abstract (摘要) 單變量c-特徵函數已被證明可處理一些難以使用傳統特徵函數解決的問題,
在本文中,我們首先提出其反演公式,透過此反演公式,我們獲得(1)Dirichlet隨機向量之線性組合的機率密度函數;(2)以一些有趣測度為參數之Ferguson-Dirichlet過程其隨機動差的機率密度函數;(3)Ferguson-Dirichlet過程之隨機泛函的Lebesgue積分表示式。

本文給予對稱分配之多變量c-特徵函數的新性質,透過這些性質,我們證明在任何$n$維球面上之Ferguson-Dirichlet過程其隨機均值是一對稱分配,並且我們亦獲得其確切的機率密度函數,此外,我們將這些結果推廣至任何n維橢球面上。

我們亦探討條件分配相容性的問題,這個問題在機率理論與貝式計算上有其重要性,我們提出其充要條件。當給定相容的條件分配時,我們不但解決相關聯合分配唯一性的問題,而且也提供方法去獲得所有可能的相關聯合分配,我們亦給予檢驗相容性、唯一性及建構機率密度函數的演算法。

透過相容性的相關理論,我們提出完整且清楚地統合性貝氏反演公式理論,並建構可應用於一般測度空間的廣義貝氏反演公式。此外,我們使用廣義貝氏反演公式提供一個配適機率密度函數的演算法,此演算法沒有疊代演算法(如Gibbs取樣法)的收斂問題。
zh_TW
dc.description.abstract (摘要) The univariate c-characteristic function has been shown to be important in cases that are hard to manage using the traditional characteristic function. In this thesis, we first give its inversion formulas. We then use them to obtain (1) the probability density functions (PDFs) of a linear combination of the components of a Dirichlet random vector; (2) the PDFs of random functionals of a Ferguson-Dirichlet process with some interesting parameter measures; (3) a Lebesgue integral expression of any random functional
of the Ferguson-Dirichlet process.

New properties of the multivariate c-characteristic function with a spherical distribution are given in this thesis. With them, we show that the random mean of a Ferguson-Dirichlet process over a spherical surface in n dimensions has a spherical distribution on the n-dimensional ball. Moreover, we derive its exact PDF. Furthermore, we generalize this result to any ellipsoidal surface in n-space.

We also study the issue of compatibility for specified conditional distributions. This issue is important in probability theory and Bayesian computations. Several necessary and sufficient conditions for the compatibility are provided. We also address the problem of uniqueness of the associated joint distribution when the given conditionals are compatible. In addition, we provide a method to obtain all possible joint distributions that have the given compatible conditionals. Algorithms for checking the compatibility and the uniqueness, and for constructing all associated densities are also given.

Through the related compatibility theorems, we provide a fully and cleanly unified theory of inverse Bayes formula (IBF) and construct a generalized IBF (GIBF) that is applicable in the more general measurable space. In addition, using the GIBF, we provide a marginal density fitting algorithm, which avoids the problems of convergence in iterative algorithm such as the Gibbs sampler.
en_US
dc.description.tableofcontents 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
zh_TW
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dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0917515011en_US
dc.subject (關鍵詞) c-特徵函數zh_TW
dc.subject (關鍵詞) 相容性zh_TW
dc.subject (關鍵詞) Ferguson-Dirichlet過程zh_TW
dc.subject (關鍵詞) 廣義貝氏反演公式zh_TW
dc.subject (關鍵詞) 隨機泛函zh_TW
dc.subject (關鍵詞) c-characteristic functionen_US
dc.subject (關鍵詞) compatibilityen_US
dc.subject (關鍵詞) Ferguson-Dirichlet processen_US
dc.subject (關鍵詞) generalized inverse Bayes formulaen_US
dc.subject (關鍵詞) random functionalen_US
dc.title (題名) 研究Ferguson-Dirichlet過程和條件分配族相容性之新工具zh_TW
dc.title (題名) New tools for studying the Ferguson-Dirichlet process and compatibility of a family of conditionalsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) Amemiya, T. (1975) Qualitative response models. Annals of Economic and Social Measurement, 4, 363-372.zh_TW
dc.relation.reference (參考文獻) Arnold, B.C. and Gokhale, D.V. (1998) Distributions of the most nearly compatible with given families of conditional distributions. Test, 7, 377-390.zh_TW
dc.relation.reference (參考文獻) Arnold, B.C. and Press, S.J. (1989) Compatible conditional distributions. J. Amer. Statist. Assoc., 84, 152-156.zh_TW
dc.relation.reference (參考文獻) Arnold, B.C., Castillo, E. and Sarabia, J.M. (2001) Conditionally specified distributions: an introductionzh_TW
dc.relation.reference (參考文獻) (with discussion). Statist. Sci., 16, 249-274.zh_TW
dc.relation.reference (參考文獻) Arnold, B.C., Castillo, E., and Sarabia, J.M. (2002) Exact and near compatibility of discrete conditional distributions. Comput. Statist. Data Anal., 40, 231-252.zh_TW
dc.relation.reference (參考文獻) Arnold, B.C., Castillo, E., and Sarabia, J.M. (2004) Compatibility of partial or complete conditional probabilityzh_TW
dc.relation.reference (參考文獻) specifications. J. Statist. Plann. Inference, 123, 133-159.zh_TW
dc.relation.reference (參考文獻) Carlson, B.C. (1977) Special Functions of Applied Mathematics. New York: Academic Press.zh_TW
dc.relation.reference (參考文獻) Casella, G. and George, E.I. (1992) Explaining the Gibbs sampler. Amer. Statist., 46, 167-174.zh_TW
dc.relation.reference (參考文獻) Chung, K.L. (1974) A Course in Probability Theory. New York: Academic Press.zh_TW
dc.relation.reference (參考文獻) Cifarelli, D.M. and Regazzini, E. (1990) Distribution functions of means of a Dirichlet process. Ann. Statist., 18, 429-442. Correction (1994): Ann. Statist., 22, 1633-1634.zh_TW
dc.relation.reference (參考文獻) Cifarelli, D.M. and Melilli, E. (2000) Some new results for Dirichlet Priors. Ann. Statist., 28, 1390-1413.zh_TW
dc.relation.reference (參考文獻) Dempster, A.P., Laird, N.M., and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B, 39, 1-38.zh_TW
dc.relation.reference (參考文獻) Diaconis, P. and Kemperman, J. (1996) Some new tools for Dirichlet priors. In J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith (eds.), Bayesian Statistics 5, pp. 97-106. Oxford University Press.zh_TW
dc.relation.reference (參考文獻) Dickey, J.M., Jiang, T.J., and Kuo, K.-L. (2008), Functionals of a Ferguson-Dirichlet process. Preprint.zh_TW
dc.relation.reference (參考文獻) Epifani, I., Guglielmi, A., and Melilli, E. (2006) A stochastic equation for the law of the random Dirichlet variance. Statist. Probab. Lett., 76 , 495-502.zh_TW
dc.relation.reference (參考文獻) Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1953) Higher Transcendental Functions, vol. I. New York: McGraw-Hill.zh_TW
dc.relation.reference (參考文獻) Ferguson, T.S. (1973) A Bayesian analysis of some nonparametric problems. Ann. Statist., 1, 209-230.zh_TW
dc.relation.reference (參考文獻) Gelfand, A.E. and Smith, A.F.M. (1990) Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc., 85, 398-409.zh_TW
dc.relation.reference (參考文獻) Gourieroux, C. and Montfort, A. (1979) On the characterization of a joint probability distribution by conditional distributions. J. Econometrics, 10, 115-118.zh_TW
dc.relation.reference (參考文獻) Gradshteyn, I.S. and Ryzhik, I.M. (2000) Table of Integrals, Series, and Products, 6th ed. New York: Academic Press.zh_TW
dc.relation.reference (參考文獻) Grobner, W. and Hofreiter, W. (1973) Integraltafel, Vol. 2, 5th ed. New York: Springer-Verlag.zh_TW
dc.relation.reference (參考文獻) Hannum, R.C., Hollander, M., and Langberg, N.A. (1981) Distributional results for random functionals of a Dirichlet process. Ann. Probab., 9, 665-670.zh_TW
dc.relation.reference (參考文獻) Hjort, N.L. and Ongaro, A. (2005) Exact inference for random Dirichlet means. Stat. Inference Stoch. Process., 8, 227-254.zh_TW
dc.relation.reference (參考文獻) Jiang, J. (1988) Starlike functions and linear functions of a Dirichlet distributed vector. SIAM J. Math. Anal., 19, 390-397.zh_TW
dc.relation.reference (參考文獻) Jiang, T.J. (1991) Distribution of random functional of a Dirichlet process on the unit disk. Statist. Probab. Lett., 12, 263-265.zh_TW
dc.relation.reference (參考文獻) Jiang, T.J., Dickey, J.M., and Kuo, K.-L. (2004) A new multivariate transform and the distribution of azh_TW
dc.relation.reference (參考文獻) random functional of a Ferguson-Dirichlet process.zh_TW
dc.relation.reference (參考文獻) Stochastic Process. Appl., 111, 77-95.zh_TW
dc.relation.reference (參考文獻) Jiang, T.J. and Kuo, K.-L. (2008a), Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere. To appear in Electron. Comm. Probab..zh_TW
dc.relation.reference (參考文獻) Jiang, T.J. and Kuo, K.-L. (2008b), The inversion formula of the c-characteristic function and its applications.zh_TW
dc.relation.reference (參考文獻) Preprint.zh_TW
dc.relation.reference (參考文獻) Kuo, K.-L. (2002) Some applications of multivariate c-transformations. Master thesis, Department of Mathematical Sciences, National Chengchi University.zh_TW
dc.relation.reference (參考文獻) Kuo, K.-L., Song, C.-C., and Jiang, T.J. (2008a),zh_TW
dc.relation.reference (參考文獻) Compatibility of discrete conditionals in higher dimensions. Preprint.zh_TW
dc.relation.reference (參考文獻) Kuo, K.-L., Song, C.-C., and Jiang, T.J. (2008b), Compatible continuous conditionals and an application on normal conditionals. Preprint.zh_TW
dc.relation.reference (參考文獻) Kuo, K.-L., Song, C.-C., and Jiang, T.J. (2008c), Generalized inverse Bayes formula for compatible conditional distributions. Preprint.zh_TW
dc.relation.reference (參考文獻) Lijoi, A. and Regazzini, E. (2004) Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab., 32, 1469-1495.zh_TW
dc.relation.reference (參考文獻) Liu, J.S. (1996) Discussion of "Statistical Inference and Monte Carlo Algorithms" by G. Casella. Test, 5, 305-310.zh_TW
dc.relation.reference (參考文獻) Lord, R.D. (1954) The use of the Hankel transformations in statistics. I. General theory and example. Biometrika, 41, 44-55.zh_TW
dc.relation.reference (參考文獻) Minc, H. (1988) Nonnegative Matrices. New York: Wiley.zh_TW
dc.relation.reference (參考文獻) Nerlove, M. and Press, S.J. (1986) Multivariate log-linear probability models in econometrics. In Advances in Statistical Analysis and Statistical Computing (Edited by Mariano, R. S.), 117-171. Greenwich, CT: JAI Press.zh_TW
dc.relation.reference (參考文獻) Ng, K.W. (1995) Explict formulas for unconditional pdf.zh_TW
dc.relation.reference (參考文獻) Research Report, No. 82 (revised). Department of Statistics, University of Hong Kong.zh_TW
dc.relation.reference (參考文獻) Ng, K.W. (1997) Inversion of Bayes formula: explict formulaezh_TW
dc.relation.reference (參考文獻) for unconditional pdf. In Advance in the Theory and Practice in Statistics (Edited by Johnson, N. L. and Balakrishnan, N.), 571-584, New York: Wiley.zh_TW
dc.relation.reference (參考文獻) Perez-Villalta, R. (2000) Variables finitas condicionalmente especificadas. Questioo, 24, 425-448.zh_TW
dc.relation.reference (參考文獻) Provost, S.B. and Cheong, Y.-H. (2000) On the distribution of linear combinations of the components of a Dirichlet random vector. Canad. J. Statist., 28, 417-425.zh_TW
dc.relation.reference (參考文獻) Prudnikov, A.P., Brychkov, Yu.A., and Marichev, O.I. (1986)zh_TW
dc.relation.reference (參考文獻) Integrals and Series, Vol. 3. New York: Gordon and Breach Science Publishers.zh_TW
dc.relation.reference (參考文獻) Rao, C.R. (1973) Linear Statistical Inference and Its Applications, New York: Wiley.zh_TW
dc.relation.reference (參考文獻) Regazzini, E., Guglielmi, A., and Di Nunno, G. (2002)zh_TW
dc.relation.reference (參考文獻) Theory and numerical analysis for exact distributions ofzh_TW
dc.relation.reference (參考文獻) functionals of a Dirichlet process. Ann. Statist., 30, 1376-1411.zh_TW
dc.relation.reference (參考文獻) Song, C.-C., Li, L.-A., Chen, C.-H., Jiang, T.J., and Kuo, K.-L. (2006), Compatibility of finite discrete conditional distributions. Under revision.zh_TW
dc.relation.reference (參考文獻) Sumner, D.B. (1949) An inversion formula for the generalized Stieltjes transform. Bull. Amer. Math. Soc., 55, 174-183.zh_TW
dc.relation.reference (參考文獻) Tan, M., Tian, G.-L. and Ng, K. W. (2003).A noniterative sampling method for computing posteriors in the structure of EM-type algorithms. Statist. Sinica, 13, 625-639.zh_TW
dc.relation.reference (參考文獻) Tanner, M.A. and Wong, W.H. (1987) The calculation of posterior distributions by data augmentation (with discussion). J. Amer. Statist. Assoc., 82, 528-540.zh_TW
dc.relation.reference (參考文獻) Tian, G.-L., Ng, K.W., and Geng, Z. (2003) Bayesian computation for contingency tables with incomplete cell-counts. Statist. Sinica, 13, 189-206.zh_TW
dc.relation.reference (參考文獻) Tian, G.-L. and Tan, M. (2003) Exact statistical solutions using the inversion Bayes formulae. Statist. Probab. Lett., 62, 305-315.zh_TW
dc.relation.reference (參考文獻) Tian, G.-L., Tan, M. and Ng, K.W. (2007) An exact non-iterative sampling procedure for discrete missing data problems. Statist. Neerlandica, 61, 232-242.zh_TW
dc.relation.reference (參考文獻) Weisstein, E.W. (2005) Permutation Matrix. From MathWorld-A Wolfram Web Resource.zh_TW
dc.relation.reference (參考文獻) http://mathworld.wolfram.com/PermutationMatrix.htmlzh_TW
dc.relation.reference (參考文獻) Widder, D.V. (1946) The Laplace Transform. Princenton University Press.zh_TW
dc.relation.reference (參考文獻) Yamato, H. (1984) Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab., 12, 262-267.zh_TW
dc.relation.reference (參考文獻) Zayed, A.I. (1996) Handbook of function and generalized function transformations. New York: CRC Press.zh_TW