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System No. U0002-2106200605554200
Title (in Chinese) 一些二元機率結合函數的探討
Title (in English) On Some Bivariate Copulas
Other Title
Institution 淡江大學
Department (in Chinese) 管理科學研究所碩士班
Department (in English) Graduate Institute of Management Science
Other Division
Other Division Name
Other Department/Institution
Academic Year 94
Semester 2
PublicationYear 95
Author's name (in Chinese) 吳文獻
Author's name(in English) Wen-Hsien Wu
Student ID 693560277
Degree 碩士
Language Traditional Chinese
Other Language
Date of Oral Defense 2006-06-02
Pagination 30page
Committee Member advisor - Wen-Tao Huang
co-chair - 周青松
co-chair - 婁國仁
Keyword (inChinese) 二元機率結合函數
Keyword (in English) Bivariate Copulas
Bivariate Distribution
Other Keywords
Abstract (in Chinese)
在第一章,先回顧目前一些二元分佈和二元機率結合函數(Bivariate Copulas)相關的定義、主要定理,及目前的發展概況。接著介紹一些常見的機率結合函數,如AC族、FGM族及常態機率結合函數族。由回顧知道,經由二元機率合函數以建立二元分佈是一項很有用的工具,尤其是邊際分佈為特定分佈時,如邊際分佈為同類型時之二元指數、二元迦瑪等等。
在第三章,經觀察Farlie(1960)及RULF族,領悟出一些新的二元機率結合函數族,但這些函數不屬於AC族、RULF族及目前一些已知的機率結合函數族。這些新的機率結合函數族,參數在某些特定值時,有正象相依(Positive Quadrant Dependence,PQD)的性質。
Abstract (in English)
In chapter one, we review recent developments on bivariate distributions and bivariate copulas. We also introduce some bivariate copulas, such as Archimedean copulas, Farlie-Gumbel-Morgenstern copulas  and Normal copulas . It is obvious to see that bivariate copulas is a very useful tool for constructing bivariate distributions, especially when the marginal distributions are given of the same type of distribution such as bivariate exponential distribution and bivariate gamma distribution.
In chapter two, we provide some new copulas which may be considered as some extension of RULF and this new copulas increases |ρs| and also increases its applicability for real data. We give some bivariate exponential distributions which are constructed by the proposed bivariate copulas.
Finally, in chapter three, we propose some new bivariate copulas which don’t belong to AC family nor RULF family nor other family so far it is known. Those proposed copulas possess some property of PQD.
Other Abstract
Table of Content (with Page Number)
第一章  二元分佈介紹	1
1.1  緒論	1
1.2  二元分佈相關簡介與定義	3
1.2A  機率結合函數族群	3
1.2B  FGM(Farlie-Gumbel-Morgenstern)族	8
第二章  一些二元機率結合函數族之推廣	13
2.1  二元RLUF族機率結合函數之推廣	13
第三章  一些新的二元機率結合函數族	22
3.1  由Farlie(1960)所觀得之函數族	25
3.2  由RLUF所觀得之函數族	25
附錄一  RLUF族之參數估計	27

表  圖  目  錄
表2.1 定理2.3中(1)之特殊函數	15
表2.2 定理2.3中(2)之特殊函數	16
表3.1 當θ=1-e及θ=e,(3.3)式之一階動差及Spearman相關係數	23
表3.2 當θ=-1及θ=1,(3.5)式之一階動差及Spearman相關係數	23
圖3.1  θ=e時,(3.3)之機率密度函數圖	24
圖3.2  θ=1時,(3.5)之機率密度函數圖	24
圖3.3  θ=1時,(3.10)之機率密度函數圖
[1] Alfonsi, A. and Brigo, D. (2005), “New Families of Copulas Based on Periodic Functions”, Communications in Statistics-Theory and Methods, 34, 1437-1447. 
[2] Ali, M.M., Mikhail, N.N., and Haq, M.S. (1978), “A Class of Bivariate Distributions Including the Bivariate Logistic”, Journal of Multivariate Analysis 8, 405-412.
[3] Amblard, C. and Girard, S. (2002), “Symmetry and Dependence Properties within A Semiparametric Family of Bivariate Copulas ”, Nonparametric Statistics, Vol. 14, 6, 715-727.
[4] Bairamov, I., and Kotz, S. (2002), “Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions”, Metrica, 56, 55-72.
[5] Bairamov, I.,  Kotz, S. and Bekci, M. (2001), “New generalized Farlie-Gumbel-Morgenstern distribution and concomitants of order statistics”, Journal of Applied Statistics, Vol. 28, No. 5, 521-536.
[6] Clayton, D. and Cuzick, J. (1985), “Multivariate Generalizations of the Proportional Hazards Model”, Journal of the Royal Statistical Society Series A, 148, 82-117.
[7] Farlie, D.J.G. (1960), “The Performance of Some correlation coefficients for a general Bivariate distribution”, Biometrika, 47, 307-233.
[8] Finkelstein, M.S. (2003), “On One Class of Bivariate Distributions”, Statistics & Probability Letters, 65, 1-6.
[9] Fisher, N.I. (1997), “Copulas”,in Encyclopedia of Statistical Sciences, Update Vol. 1, 159-163, John Wiley & Son.
[10] Gumbel, E.J. , “Bivariate Exponential Distributions”, Journal of the American Statistical Association, Vol. 55, No. 292, 698-707.
[11] Hu, L.(2003), “Dependence Patterns Across Financial Markets: a Mixed Copulas Approach”, OSU.
[12] Hung, J.S. and Kotz, S., (1999), “Modifications of the Farlie-Gumbel-Morgenstern distributions. A tough hill to climb” ,Metrika, 49, 135-145.
[13] Johnson, N.L. and Kotz, S.(1975) , “On Some Generalized Farlie-Gumbel-Morgenstern Distributions ”,Communication in Statistics, 4, 415-427.
[14] Johnson, N.L. and Kotz, S.(1977) , “On Some Generalized Farlie-Gumbel-Morgenstern Distributions II: Regression, Correlation An Further Generalizations ”,Communication in Statistics, 6, 485-496.
[15] Kotz, S., Balakrishnan, N.,  and  Johnson, N. S. (2000), “Continuous  Multivariate Distribution”, Vol. 1, second edition, John Wiley & Son.
[16] Lai, C.D. and Xie M. (2000), “A New Family of Positive Quadrant Dependent Bivariate Distribution”, Statistics & Probability Letters, 46, 359-364.
[17] Li, D.X. (1999), “On default correlation: a copula function approach”, The Journal of Fixed Income, 9, 43-54.
[18] Morgenstern, D. (1956) , “Einfache Beispiele zweidimensionaler Verteilungen”, Mitteilungsblatt fur Mathmatische Statistik, 8, 234-235.
[19] Nelsen, R.B. (1999), “An Introduction to Copulas”, Springer, New York.
[20] Rodriguez-Lallena, J.A. and Úbeda-Flores, M. (2004), “A New Class of Bivariate Copulas”, Statistics & Probability Letters, 66, 315-325.
[21] Rohatgi, V.K. (1976), “An Introduction to Probability and Mathmatical Statistics”, John Wiley & Son.
[22] Sklar, A. (1959), “Fonctions de reparition a n dimensions et leurs marges”,Publications de 1’Institut de Statistique de 1’Universite de Paris, 8, 229-231.
[23] Sungur, E.A. (2005), “Some Observations on Copula Regression Functions”, Communications in Statistics-Theory and Methods, 34, 1976-1978.
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