雙變量Gamma與廣義Gamma分配之探討｜國立政治大學博碩士論文系統

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研究生：	曾奕翔
論文名稱：	雙變量Gamma與廣義Gamma分配之探討
指導教授：	陳麗霞
學位類別：	博士 Doctor
系所名稱：	商學院 - 統計學系 Department of Statistics
論文出版年：	2009
畢業學年度：	97
語文別：	中文
論文頁數：	141
中文關鍵詞：	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
外文關鍵詞：	bivariate generalized gamma distribution, bivariate normal distribution, survival analysis, sensitivity analysis
相關次數：	點閱：368 下載：158
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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.

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

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.

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