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題名 雙變量Gamma與廣義Gamma分配之探討
作者 曾奕翔
貢獻者 陳麗霞
曾奕翔
關鍵詞 雙變量廣義伽瑪分配
雙變量常態分配
存活分析
敏感度分析
bivariate generalized gamma distribution
bivariate normal distribution
survival analysis
sensitivity analysis
日期 2009
上傳時間 9-May-2016 15:11:43 (UTC+8)
摘要 Stacy (1962)首先提出廣義伽瑪分配 (generalized gamma distribution)，此分布被廣泛應用於存活分析 (survival analysis) 以及可靠度 (reliability) 中壽命時間的資料描述。事實上，像是指數分配 (exponential distribution)、韋伯分配 (Weibull distribution) 以及伽瑪分配 (gamma distribution) 都是廣義伽瑪分配的一個特例。
     Bologna (1987)提出一個特殊的雙變量廣義伽瑪分配 (bivariate generalized gamma distribution) 可以經由雙變量常態分配 (bivariate normal distribution) 所推得。我們根據他的想法，提出多變量廣義伽瑪分配可以經由多變量常態分配所推得。在過去的研究中，學者們做了許多有關雙變量伽瑪分配。當我們提到雙變量常態分配，由於其分配的型式為唯一的，所以沒人任何人對其分配的型式有疑問。然而，雙變量伽瑪分配卻有很多不同的型式。
     在這篇論文中的架構如下。在第二章中，我們介紹並討論雙變量廣義伽瑪分配可以經由雙變量常態分配所推得，接著推導參數估計以及介紹模擬的程序。在第三章中，我們介紹一些對稱以及非對稱的雙變量伽瑪分配，接著拓展到雙變量廣義伽瑪分配，有關參數的估計以及模擬結果也將在此章中討論。在第三章最後，我們建構參數的敏感度分析 (sensitivity analysis)。最後，在第四章中，我們陳述結論以及未來研究方向。
The generalized gamma distribution was introduced by Stacy (1962). This distribution is useful to describe lifetime data when conducting survival analysis and reliability. In fact, it includes the widely used exponential, Weibull, and gamma distributions as special cases.
      Bologna (1987) showed that a special bivariate genenralized gamma distribution can be derived from a bivariate normal distribution. Follow his idea, we show that a multivariate generalized gamma distribution can be derived from a multivariate normal distribution. In the past, researchers spend much time in working on a bivariate gamma distribution. When a bivariate normal distribution is mentioned, no one feels puzzled about its form, since it has only one form. However, there are various forms of bivariate gamma distributions.
     In this paper is as following. In Chapter 2, we introduce and discuss the bivariate generalized gamma distribution, then the multivariate generalized gamma distribution is derived. We also develop parameters estimation and simulation procedure. In Chapter 3, we introduce some symmetrical and asymmetrical bivariate gamma distributions, then they are extended to the bivariate generalized gamma distributions. Problems of parameters estimation and simulation results are also discussed in Chapter 3. Besides, sensitivity analyses of parameters estimation are conducted. Finally, we state conclusion and future work in Chapter 4.
參考文獻 Bologna, S. (1987). On a bivariate generalized gamma distribution. Statistica, 47, 543-548.
     
     Bithas, P. S., Nikos C. S., Theodoros A. T. and George K. K. (2007). Products and ratios of two Gaussian class correlated Weibull random variables, in The 12th International Conference, ASMDA 2007. (Channia, Greece).
     
     Bithas, P. S., Nikos C. S., Theodoros A. T. and George K. K. (2007). Distributions involving correlated generalized gamma variables, in The 12th International Conference, ASMDA 2007. (Channia, Greece).
     
     Chatelain, F., Tourneret, J.-Y., Inglada, J. and Ferrari, A. (2006). Parameter estimation for multivariate gamma distributions. Application to image registration, in Proc. EUSIPCO-06, (Florence, Italy).
     
     Chatelain, F. and Tourneret, J.-Y. (2007). Bivariate gamma distributions for multisensor sar images, in Acoustics, Speech and Signal Processing, ICASSP 2007. IEEE International Conference.
     
     Cherian, K. C. (1941). A bivariate correlated gamma-type distribution function. Journal of the Indian Mathematical Society, 5, 133-144.
     
     Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products. Academic Press, New York.
     
     Hardy, G.. H. (1932). Summation of a series of polynomials of Laguerre. Journal of the London Mathematical Society, 8, 138-139.
     
     Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
     
     Joe, H. and Xu, J. (1996). The estimation method of inference functions for margins for multivariate models. Department of Statistics, University of British Columbia. Technical Report, 166.
     
     Kibble, W. F. (1941). A two-variate gamma type distribution. Sankhya, 5, 137-150.
     Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000). Continuous Multivariate Distributions. Second Edition. New York: John Wiley & Sons.
     
     Loaiciga, H. A. and Leipink, R. B. (2005). Correlated gamma variables in the analysis of microbial densities in water. Advance in Water Resource, 28, 329-335.
     
     Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics, 33, 409-419.
     
     Mckay, A. T. (1934). Sampling from batches. Journal of the Royal Statistical Society, 1, 207-216.
     
     Mardia, K. V. (1970). Families of Bivariate Distributions. Griffin, London
     
     Moran, P. A. P. (1967). Testing for correlation between non-negative variates. Biometrika, 54, 385-394.
     
     Nadarajah, S. and Gupta, A. K. (2006). Some bivariate gamma distributions. Applied Mathematics Letters, 19, 767-774.
     
     Nadarajah, S. and Kotz, S. (2006). Bivariate gamma distributions, sums, and ratios. Bulletin of the Brazilian Mathematical Society, New Series, 37(2), 241-274.
     
     Nadarajah, S. and Kotz, S. (2007). A note on the correlated gamma distribution of Loaiciga and Leipnik. Advance in Water Resource, 30, 1053-1055.
     
     Nestler, J. M. (1993). Instream flow incremental methodology: A synopsis with recommendations for use and suggestions for future research. Technical Report, EL-93-3.
     
     Piboongungon, T., Aalo, V. A., Iskander, C. D. and Efthymoglou, G. P. (2005). Bivariate generalised gamma distribution with arbitrary fading parameters. Electronics Letters, 41, 709-710.
     
     Ramabhadran, V. R. (1951). A multivariate gamma-type distribution. Sankhya, 11, 45-46.
     
     Sarmanov, I. O. (1970). Gamma correlation process and its properties. Doklady Akademii Nauk, SSSR, 191, 30-32.
     
     Smith, O. E., Adelfang, S. I. and Tubbs, J. D. (1982). A bivariate gamma probability distribution with application to guest modeling. NASA Technical Report TM-82483.
     
     Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics, 33, 1187-1192.
     
     Tunaru, R. and Albota, G. (2005). Estimating risk neutral density with a generalized gamma distribution. Cass Business School Working Paper.
     
     Wolfram, S (2006). The Wolfram functions site. Internet, http://integrals.wolfram.com
     
     Yue, S., Ouarda, T.B.M.J. and Bobee, B. (2001). A review of bivariate gamma distributions for hydrological application. Journal of Hydrology, 246, 1-18.
描述 博士
國立政治大學
統計學系
91354502
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0913545023
資料類型 thesis
dc.contributor.advisor 陳麗霞zh_TW
dc.contributor.author (Authors) 曾奕翔zh_TW
dc.creator (作者) 曾奕翔zh_TW
dc.date (日期) 2009en_US
dc.date.accessioned 9-May-2016 15:11:43 (UTC+8)-
dc.date.available 9-May-2016 15:11:43 (UTC+8)-
dc.date.issued (上傳時間) 9-May-2016 15:11:43 (UTC+8)-
dc.identifier (Other Identifiers) G0913545023en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/95125-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 91354502zh_TW
dc.description.abstract (摘要) Stacy (1962)首先提出廣義伽瑪分配 (generalized gamma distribution)，此分布被廣泛應用於存活分析 (survival analysis) 以及可靠度 (reliability) 中壽命時間的資料描述。事實上，像是指數分配 (exponential distribution)、韋伯分配 (Weibull distribution) 以及伽瑪分配 (gamma distribution) 都是廣義伽瑪分配的一個特例。
     Bologna (1987)提出一個特殊的雙變量廣義伽瑪分配 (bivariate generalized gamma distribution) 可以經由雙變量常態分配 (bivariate normal distribution) 所推得。我們根據他的想法，提出多變量廣義伽瑪分配可以經由多變量常態分配所推得。在過去的研究中，學者們做了許多有關雙變量伽瑪分配。當我們提到雙變量常態分配，由於其分配的型式為唯一的，所以沒人任何人對其分配的型式有疑問。然而，雙變量伽瑪分配卻有很多不同的型式。
     在這篇論文中的架構如下。在第二章中，我們介紹並討論雙變量廣義伽瑪分配可以經由雙變量常態分配所推得，接著推導參數估計以及介紹模擬的程序。在第三章中，我們介紹一些對稱以及非對稱的雙變量伽瑪分配，接著拓展到雙變量廣義伽瑪分配，有關參數的估計以及模擬結果也將在此章中討論。在第三章最後，我們建構參數的敏感度分析 (sensitivity analysis)。最後，在第四章中，我們陳述結論以及未來研究方向。		zh_TW
dc.description.abstract (摘要) The generalized gamma distribution was introduced by Stacy (1962). This distribution is useful to describe lifetime data when conducting survival analysis and reliability. In fact, it includes the widely used exponential, Weibull, and gamma distributions as special cases.
      Bologna (1987) showed that a special bivariate genenralized gamma distribution can be derived from a bivariate normal distribution. Follow his idea, we show that a multivariate generalized gamma distribution can be derived from a multivariate normal distribution. In the past, researchers spend much time in working on a bivariate gamma distribution. When a bivariate normal distribution is mentioned, no one feels puzzled about its form, since it has only one form. However, there are various forms of bivariate gamma distributions.
     In this paper is as following. In Chapter 2, we introduce and discuss the bivariate generalized gamma distribution, then the multivariate generalized gamma distribution is derived. We also develop parameters estimation and simulation procedure. In Chapter 3, we introduce some symmetrical and asymmetrical bivariate gamma distributions, then they are extended to the bivariate generalized gamma distributions. Problems of parameters estimation and simulation results are also discussed in Chapter 3. Besides, sensitivity analyses of parameters estimation are conducted. Finally, we state conclusion and future work in Chapter 4.		en_US
dc.description.tableofcontents Chapter 1 Introduction 1
     Chapter 2 Multivariate Generalized Gamma Distribution Derived from Multivariate Normal Distribution 5
     2.1 Bivariate Generalized Gamma Distribution Derived from Bivariate Normal
     Distribution 5
     2.2 Multivariate Generalized Gamma Distribution 18
     2.3 Parameters Estimation 19
     2.3.1 Method of Moments 20
     2.3.2 Maximum Likelihood Method 21
     2.3.3 Inference Function for Margins Method 25
     2.4 Simulation Results 28
     2.4.1 Simulation Procedure 28
     2.4.2 Performance 28
     Chapter 3 Bivariate Generalized Gamma Distributions Derived from Bivariate Gamma Distributions 36
     3.1 Symmetrical and Unsymmetrical Bivariate Gamma Distributions 37
     3.2 Bivariate Generalized Gamma Distribution 45
     3.3 Parameters Estimation for Unsymmetrical Bivariate Generalized Gamma Distribution 81
     3.3.1 Method of Moments 81
     3.3.2 Maximum Likelihood Method 83
     3.3.3 Inference Function for Margins Method 87
     3.4 Simulation Results 89
     3.4.1 Simulation Procedure 89
     3.4.2 Performance 91
     3.5 Sensitivity Analysis 95
     3.5.1 Simulation Procedure 95
     Chapter 4 Conclusion and Future Work 125
     References 128
     Appendix A：Special Functions and Notations 131
     Appendix B：Mathematical Formulas 132
     Appendix C：Simulation Programs 133
     Table 2.1 29
     Table 2.2 29
     Table 2.3 29
     Table 2.4 29
     Table 2.5 30
     Table 2.6 30
     Table 2.7 30
     Table 2.8 30
     Table 2.9 30
     Table 2.10 30
     Table 3.1 91
     Table 3.2 91
     Table 3.3 91
     Table 3.4 91
     Table 3.5 92
     Table 3.6 92
     Table 3.7 92
     Table 3.8 92
     Table 3.9 92
     Table 3.10 92
     Table 3.11 97
     Table 3.12 98
     Table 3.13 99
     Table 3.14 100
     Table 3.15 101
     Table 3.16 102
     Table 3.17 103
     Table 3.18 104
     Table 3.19 105
     Table 3.20 106
     Table 3.21 107
     Table 3.22 108
     Table 3.23 109
     Table 3.24 110
     Figure 2.1 7
     Figure 2.2 7
     Figure 2.3 31
     Figure 2.4 31
     Figure 2.5 32
     Figure 2.6 33
     Figure 2.7 33
     Figure 2.8 34
     Figure 2.9 34
     Figure 2.10 35
     Figure 3.1 57
     Figure 3.2 57
     Figure 3.3 93
     Figure 3.4 93
     Figure 3.5 94
     Figure 3.6 111
     Figure 3.7 112
     Figure 3.8 113
     Figure 3.9 114
     Figure 3.10 115
     Figure 3.11 116
     Figure 3.12 117
     Figure 3.13 118
     Figure 3.14 119
     Figure 3.15 120
     Figure 3.16 121
     Figure 3.17 122
     Figure 3.18 123
     Figure 3.19 124		zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0913545023en_US
dc.subject (關鍵詞) 雙變量廣義伽瑪分配zh_TW
dc.subject (關鍵詞) 雙變量常態分配zh_TW
dc.subject (關鍵詞) 存活分析zh_TW
dc.subject (關鍵詞) 敏感度分析zh_TW
dc.subject (關鍵詞) bivariate generalized gamma distributionen_US
dc.subject (關鍵詞) bivariate normal distributionen_US
dc.subject (關鍵詞) survival analysisen_US
dc.subject (關鍵詞) sensitivity analysisen_US
dc.title (題名) 雙變量Gamma與廣義Gamma分配之探討zh_TW
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Bologna, S. (1987). On a bivariate generalized gamma distribution. Statistica, 47, 543-548.
     
     Bithas, P. S., Nikos C. S., Theodoros A. T. and George K. K. (2007). Products and ratios of two Gaussian class correlated Weibull random variables, in The 12th International Conference, ASMDA 2007. (Channia, Greece).
     
     Bithas, P. S., Nikos C. S., Theodoros A. T. and George K. K. (2007). Distributions involving correlated generalized gamma variables, in The 12th International Conference, ASMDA 2007. (Channia, Greece).
     
     Chatelain, F., Tourneret, J.-Y., Inglada, J. and Ferrari, A. (2006). Parameter estimation for multivariate gamma distributions. Application to image registration, in Proc. EUSIPCO-06, (Florence, Italy).
     
     Chatelain, F. and Tourneret, J.-Y. (2007). Bivariate gamma distributions for multisensor sar images, in Acoustics, Speech and Signal Processing, ICASSP 2007. IEEE International Conference.
     
     Cherian, K. C. (1941). A bivariate correlated gamma-type distribution function. Journal of the Indian Mathematical Society, 5, 133-144.
     
     Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products. Academic Press, New York.
     
     Hardy, G.. H. (1932). Summation of a series of polynomials of Laguerre. Journal of the London Mathematical Society, 8, 138-139.
     
     Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
     
     Joe, H. and Xu, J. (1996). The estimation method of inference functions for margins for multivariate models. Department of Statistics, University of British Columbia. Technical Report, 166.
     
     Kibble, W. F. (1941). A two-variate gamma type distribution. Sankhya, 5, 137-150.
     Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000). Continuous Multivariate Distributions. Second Edition. New York: John Wiley & Sons.
     
     Loaiciga, H. A. and Leipink, R. B. (2005). Correlated gamma variables in the analysis of microbial densities in water. Advance in Water Resource, 28, 329-335.
     
     Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics, 33, 409-419.
     
     Mckay, A. T. (1934). Sampling from batches. Journal of the Royal Statistical Society, 1, 207-216.
     
     Mardia, K. V. (1970). Families of Bivariate Distributions. Griffin, London
     
     Moran, P. A. P. (1967). Testing for correlation between non-negative variates. Biometrika, 54, 385-394.
     
     Nadarajah, S. and Gupta, A. K. (2006). Some bivariate gamma distributions. Applied Mathematics Letters, 19, 767-774.
     
     Nadarajah, S. and Kotz, S. (2006). Bivariate gamma distributions, sums, and ratios. Bulletin of the Brazilian Mathematical Society, New Series, 37(2), 241-274.
     
     Nadarajah, S. and Kotz, S. (2007). A note on the correlated gamma distribution of Loaiciga and Leipnik. Advance in Water Resource, 30, 1053-1055.
     
     Nestler, J. M. (1993). Instream flow incremental methodology: A synopsis with recommendations for use and suggestions for future research. Technical Report, EL-93-3.
     
     Piboongungon, T., Aalo, V. A., Iskander, C. D. and Efthymoglou, G. P. (2005). Bivariate generalised gamma distribution with arbitrary fading parameters. Electronics Letters, 41, 709-710.
     
     Ramabhadran, V. R. (1951). A multivariate gamma-type distribution. Sankhya, 11, 45-46.
     
     Sarmanov, I. O. (1970). Gamma correlation process and its properties. Doklady Akademii Nauk, SSSR, 191, 30-32.
     
     Smith, O. E., Adelfang, S. I. and Tubbs, J. D. (1982). A bivariate gamma probability distribution with application to guest modeling. NASA Technical Report TM-82483.
     
     Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics, 33, 1187-1192.
     
     Tunaru, R. and Albota, G. (2005). Estimating risk neutral density with a generalized gamma distribution. Cass Business School Working Paper.
     
     Wolfram, S (2006). The Wolfram functions site. Internet, http://integrals.wolfram.com
     
     Yue, S., Ouarda, T.B.M.J. and Bobee, B. (2001). A review of bivariate gamma distributions for hydrological application. Journal of Hydrology, 246, 1-18.		zh_TW