臺灣博碩士論文加值系統

自從Azzalini(1985,1986)提出關於偏斜常態分佈的一些基本的性質開始，就有許多基於某些常見對稱分佈的偏斜分佈之研究出現。這類偏斜-對稱分佈不僅包含原本的對稱分佈，往往也有一些與原本對稱分佈相同的性質。
在本論文裡，我們考慮兩個有關偏斜-對稱分佈的主題。在第一章中，我們研究多變量偏斜常態-對稱分佈的二次形式。基於Gupta and Chang(2003)這篇論文，我們推導出多變量偏斜常態-對稱分佈，即考慮機率密度函數具有下列形式f_Z(z)=2 phi_p(z;Omega)G(alpha'z)，其中Omega> 0, alpha in R^p，phi_p(z;Omega)為一p維常態分佈的機率密度函數, 期望向量為0，且相關係數矩陣為Omega>0, 而G為滿足G'對稱於0的絕對連續分佈函數。我們先推導出某些二次形式的動差母函數，並發現一些二次形式的分佈與G獨立。其次我們推導出一個線性組合與一個二次形式，及兩個二次形式的聯合動差母函數, 並且給出一些它們相互獨立的條件。最後我們將G分別取為常態、雙指數、logistic及均勻分佈的分佈函數，來看其各自的一些特別的二次形式之分佈。
在第二章中，我們研究的是單變量的廣義偏斜-柯西分佈。我們先推導廣義偏斜-對稱分佈。假設Y是一絕對連續且對稱於0的隨機變數，其機率密度函數為f, 而分佈函數為F。若一隨機變數X滿足X^2\stackrel{d}{=}Y^2，則X被稱為一F(or f)之廣義偏斜分佈。接下來我們考慮廣義偏斜-柯西分佈，並給一些此分佈之特別的例子。其中有些例子是經由廣義偏斜-常態分佈或廣義偏斜-t分佈所產生的。

Since Azzalini (1985,1986) introduced the fundamental properties of the skew-normal distribution, there are many investigations about the skew distributions based on certain symmetric probability density functions. These classes of the skew-symmetric distributions include the original symmetric distribution and have some properties like the original one and yet is skew.
In this thesis, we consider two topics of the skew-symmetric models.In Chapter 1, we study the quadratic forms of multivariate skew normal-symmetric distributions. Following the paper by Gupta and Chang (2003) we generalize a multivariate skew normal-symmetric distribution with p.d.f. of the form
f_Z(z)=2 phi_p(z;Omega)G(alpha'z), where Omega> 0, alpha in R^p, phi_p(z;Omega) is the p-dimensional normal p.d.f. with zero mean vector and correlation matrix Omega, and G is taken to be an absolutely continuous distribution function such that G' is symmetric about 0. First we obtain the moment generating function of certain quadratic forms. It is interesting to find that the distributions of some quadratic forms are independent of G. Then the joint moment generating
functions of a linear compound and a quadratic form, and two quadratic forms, and conditions for their independence are given. Finally we take G to be one of normal, Laplace, logistic or uniform distribution, and determine the distribution of a special quadratic form for each case.
In Chapter 2, we study the generalized skew-Cauchy distributions. We
investigate the generalized skew-symmetric distributions. Suppose Y is an absolutely continuous random variable symmetric about 0 with probability density function f and cumulative distribution function F. If a random variable X satisfies X^2\stackrel{d}{=}Y^2, then X is said to have a generalized skew
distribution of F (or f). The generalized skew-Cauchy (GSC) distribution are considered and special examples of GSC distribution are presented. Some of these examples are generated from generalized skew-normal or generalized skew-t distributions.

Chapter 1. Quadratic Forms of Multivariate Skew Normal-Symmetric Distributions
1.1 Introduction
1.2 Moment generating functions of certain quadratic forms
1.3 Independence of linear forms and quadratic forms
1.4 Multivariate skew normal-normal model
1.4.1 M.G.F. of (Z-a)'A(Z-a)
1.5 Multivariate skew normal-Laplace model
1.5.1 M.G.F. of (Z-a)'A(Z-a)
1.6 Multivariate skew normal-logistic model
1.6.1 M.G.F. of (Z-a)'A(Z-a)
1.7 Multivariate skew normal-uniform model
1.7.1 M.G.F. of (Z-a)'A(Z-a)
Chapter 2. Generalized Skew-Cauchy Distribution
1.1 Introduction
1.2 Generalized skew distributions
1.3 The GSC models
1.4 More examples of GSC distribution
1.5 Some figures of the p.d.f. of the GSC distribution
References
小傳

1. Arellano-Valle, R.B., Gomez, H.W. and Quintana, F.A. (2004). A new class of skew-normal distributions. Commun. Statist.-Theory Meth. 33, 1975-1991.
2. Arnold, B.C. and Beaver, R.J. (2000). The skew-Cauchy distribution. Statist. & Prob. Lett. 49, 285-290.
3. Arnold, B.C. and Lin, G.-D. (2004). Characterizations of the skew-normal and generalized chi distributions. Sankhya 66, 593-606.
4. Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12, 171-178.
5. Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica 46, 199-208.
6. Azzalini, A. and Capitanio, A. (1999). Statistical applications of the
multivariate skew normal distribution. J. Roy. Statist. Soc. B 61, 579-602.
7. Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83, 715-726.
8. Chiogna, M. (1998). Some results on the scalar skew-normal distribution. J. Ital. Statist. Soc. 7,1-13.
9. Genton, M.G., He, L. and Liu, X. (2001). Moments of
skew-normal random vectors and their quadratic forms. Statist. & Prob. Lett. 51, 319-325.
10. Gupta, A.K. (2003). Multivariate skew t-distribution. Statistics 37, 359-363.
11. Gupta, A.K. and Chang, F.-C. (2003). Multivariate skew-symmetric distributions. Appl. Math. Lett. 16, 643-646.
12. Gupta, A.K., Chang, F.-C. and Huang, W.-J. (2002). Some skew-symmetric models. Random Oper. and Stoch. Equ. 10, 133-140.
13. Gupta, A.K., Gonzalez-Farias, G., and Dominguez-Molina, J.A. (2004a). A multivariate skew normal distribution. J. Multi. Anal. 89, 181-190.
14. Gupta, A.K. and Huang, W.-J. (2002). Quadratic forms in skew normal variates. J. Math. Anal. Appl. 273, 558-564.
15. Gupta, A.K. and Kollo, T. (2003). Density expansions based on the multivariate skew normal distribution. Sankhya 65, 821-835.
16. Gupta, A.K., Nguyen, T.T. and Sanqui, J.A.T. (2004b). Characterization of the skew-normal distribution. Ann. Inst. Statist. Math. 56, 351-360.
17. Gupta, R.C. and Brown, N. (2001). Reliability studies of the skew-normal distribution and its application to a strength-stress model. Commun. Statist.-Theory Meth. 30, 2427-2445.
18. Henze, N. (1986). A probabilistic representation of the `skew-normal' distribution. Scand. J. Statist. 13, 271-275.
19. Horn, R.A. and Johnson, C.R. (1996). Matrix Analysis. Cambridge University Press, Cambridge.
20. Huang, W.-J. and Chen, Y.-H. (2006). Quadratic forms of multivariate skew normal-symmetric distributions. Statist. & Prob. Lett. 76, 871-879.
21. Huang, W.-J., Huang, Y.-N. and Lai, C.-L. (2005). Characterizations of distributions based on certain powers of random variables. J. Chinese Statist. Assoc. 43,423-432.
22. Loperfido, N. (2001). Quadratic forms of skew-normal random vectors. Statist. & Prob. Lett. 54, 381-387.
23. Loperfido, N. (2002). Statistical implications of selectively reported inferential results. Statist. & Prob. Lett. 56, 13-22.
24. Nadarajah, S. and Kotz, S. (2003). Skewed distributions generated by the normal kernel. Statist. & Prob. Lett. 65, 269-277.
25. Wang, J., Boyer, J. and Genton, M.G. (2004a). A skew-symmetric representation of multivariate distributions. Statist. Sinica 14, 1259-1270.
26. Wang, J., Boyer, J. and Genton, M.G. (2004b). A note on an equivalence between chi-square and generalized skew-normal distributions. Statist. & Prob. Lett. 66, 395-398.
27. Zacks, S. (1981). Parametric Statistical Inference. Basic Theory and Modern Approaches. Pergamon Press, Oxford.