若干矩阵分布的一致渐近正态性
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摘要
对于一元及多元分布的定义及其性质我们已经很熟悉了,在多元分析中,作统计推断时,当样本的资料阵是矩阵的形式,这就需要我们把分布的概念从向量推广到矩阵的情形,这是多元分析中的一个重要的问题.
由于矩阵分布的密度函数形式繁琐,为了便于计算,统计学家们希望找到它们渐近正态分布的条件.目前,国内外现有的文献只是证明了Wishart分布及矩阵Γ分布的一致渐近正态性,对于其它矩阵分布的渐近正态性还没有讨论.本文主要从以下几个部分来研究矩阵变量r函数和矩阵变量Beta函数之间的关系,及矩阵Beta分布一致渐近正态分布的条件.
第一章,主要介绍目前矩阵分布的研究成果,以及本文研究的前提及需要用到的一些引理.
第二章,介绍一些常用的多元分布,如多元正态分布,多元Beta分布,引入两个随机密度函数的Kullback-Leibler距离,用此来证明多元t分布的一致渐近正态分布的条件.
第三章,在给出矩阵变量Γ函数和矩阵变量Beta函数的定义后,类比于多元函数的性质,证明了他们之间的关系.
第四章,根据已经得到的矩阵r分布的定义及其一致渐近正态分布的条件.类比矩阵Γ分布,利用两个随机密度函数的Kullback-Leibler距离,我们给出了矩阵Beta分布定义,一致渐近正态的定义,并证明了一些引理,最后给出了矩阵Beta分布一致渐近正态分布的条件.
we are familiar with the nature of some one-dimensional and multivariate distributions. In the multivariate analysis, the sample data matrix is a matrix form when we make statistical inference, which requires us to promote the definition of the distribution from vector theory to the matrix case. This is an important issue of the multivariate analysis.
As the density function of the matrix-variate distribution is complicate, we want to find the conditions under which a matrix-variate distribution will approach uniformly and asymptotically a normal distribution mainly for computational ease. Currently, the recent monograph only proved the uniform asymptotic normality of the Wishart distribution and matrix-variate Gamma distributions. For other matrix-variate distributions,we have not discussed. This article is mainly to describe as the following aspects, to describe the relationship between matrix-variate Gamma functions and matrix-variate Beta functions, and the conditions of uniform asymptotic normality of matrix-variate Beta distribution.
In the first chapter, we introduce the main study results of the matrix-variate distributions, and also introduce the premise of this study and some important lemmas.
In the second chapter, we present the multivariate normal distribution and multivariate Beta distributions, introduce the Kullback-Leiber distance between the two density functions, in order to obtain the conditions of the uniform asymptotic normality of multivariate t-distribution.
In the third chapter, after giving the definition of the matrix-variate Gamma function and matrix-variate Beta function, analogy to the nature of the multivariate function, we obtain the relationship between them.
In the fourth chapter, we first introduce the definition and the the conditions of the uniform asymptotic normality of matrix-variate Gamma distributions,analogy to the matrix-variate Gamma distributions, according to the Kullback-Leiber distance between the two density functions,we give the definition of the matrix-variate Beta distributions. Finally,we give the conditions under which a Beta matrix-variate distribution will approach uniformly and asymptotically a normal distribution.
由于矩阵分布的密度函数形式繁琐,为了便于计算,统计学家们希望找到它们渐近正态分布的条件.目前,国内外现有的文献只是证明了Wishart分布及矩阵Γ分布的一致渐近正态性,对于其它矩阵分布的渐近正态性还没有讨论.本文主要从以下几个部分来研究矩阵变量r函数和矩阵变量Beta函数之间的关系,及矩阵Beta分布一致渐近正态分布的条件.
第一章,主要介绍目前矩阵分布的研究成果,以及本文研究的前提及需要用到的一些引理.
第二章,介绍一些常用的多元分布,如多元正态分布,多元Beta分布,引入两个随机密度函数的Kullback-Leibler距离,用此来证明多元t分布的一致渐近正态分布的条件.
第三章,在给出矩阵变量Γ函数和矩阵变量Beta函数的定义后,类比于多元函数的性质,证明了他们之间的关系.
第四章,根据已经得到的矩阵r分布的定义及其一致渐近正态分布的条件.类比矩阵Γ分布,利用两个随机密度函数的Kullback-Leibler距离,我们给出了矩阵Beta分布定义,一致渐近正态的定义,并证明了一些引理,最后给出了矩阵Beta分布一致渐近正态分布的条件.
we are familiar with the nature of some one-dimensional and multivariate distributions. In the multivariate analysis, the sample data matrix is a matrix form when we make statistical inference, which requires us to promote the definition of the distribution from vector theory to the matrix case. This is an important issue of the multivariate analysis.
As the density function of the matrix-variate distribution is complicate, we want to find the conditions under which a matrix-variate distribution will approach uniformly and asymptotically a normal distribution mainly for computational ease. Currently, the recent monograph only proved the uniform asymptotic normality of the Wishart distribution and matrix-variate Gamma distributions. For other matrix-variate distributions,we have not discussed. This article is mainly to describe as the following aspects, to describe the relationship between matrix-variate Gamma functions and matrix-variate Beta functions, and the conditions of uniform asymptotic normality of matrix-variate Beta distribution.
In the first chapter, we introduce the main study results of the matrix-variate distributions, and also introduce the premise of this study and some important lemmas.
In the second chapter, we present the multivariate normal distribution and multivariate Beta distributions, introduce the Kullback-Leiber distance between the two density functions, in order to obtain the conditions of the uniform asymptotic normality of multivariate t-distribution.
In the third chapter, after giving the definition of the matrix-variate Gamma function and matrix-variate Beta function, analogy to the nature of the multivariate function, we obtain the relationship between them.
In the fourth chapter, we first introduce the definition and the the conditions of the uniform asymptotic normality of matrix-variate Gamma distributions,analogy to the matrix-variate Gamma distributions, according to the Kullback-Leiber distance between the two density functions,we give the definition of the matrix-variate Beta distributions. Finally,we give the conditions under which a Beta matrix-variate distribution will approach uniformly and asymptotically a normal distribution.
引文
[1]De Waal, D.J., Distribution connected with a multivariate Beta statistic[J]. Ann. Math. Statist.1970,41(3),1091-1095.
[2]Konno, Y., The Exact moments of multivariate F and Beta distribution [J]. Japan. Statist. Soc,1980,18:123-130.
[3]Khatri, C.G.,A density free approach to the matrix variate beta distribution. Sankhya series, 1970,32(1):311-318.
[4]Mitra, S.K., A identity-free approach to matrix-variate Beta distribution[J]. Sankhya series, 1970,32(1):81-88.
[5]Bellman,R.Introduction to Matrix Analysis. [M]. New York, Springer,1970.
[6]Cramer,H.Mathematical methds of statistics[M].Princenton Univ.Press, Princenton,1946.
[7]Olkin,I.,Rubin,H.,Multivariate Beta distribution and independent property with the Wishart Distribution[J].Ann.Math. Statist.1964,35(1):261-269.
[8]Fang, B.Q., Dawid, A.P., Nonconjugate Bayesian regression on many variables[J].Statist, Planning and Inference.2003,103(1):245-261.
[9]Akimoto, Y. The uniform asymptotic normality of the Wishart Based on the K-L Infrmation [J]. J.Japan.Statist.soc.1994,24(1):73-81.
[10]Muirhead, R.J., Aspects of multivariate statistical theory[M]. Wikey, New York.1982.
[11]Apostol.T.,Calculus.Vol.1,Jlhn wiley & Sons, New York,1961.
[12]Robert G O,Shau S K. Updating Schemes,Correlation Structure,Blocking and Parameteriz for the Gibbs Samplerl [J]. J R Statist,1997,59:291-317.
[13]Reiss R D. Approximate Distributions of Order Statistics.[M]. New York,Springer,1980.
[14]张尧庭,方开泰.多元统计分析引论[M].北京：科学出版社,1982.
[15]李开灿.矩阵Γ分布的一致渐近正态性[J].数学学报,2006,49(2)：435-442.
[16]Whittaker J. Graphical Models in Applied Multivariate Statistics [M]. Wiley,Chichester,1990.
[17]Khatri,C. G. On the mutual independence of certain statistics[J]. Ann. Math. Statist.1959, 30(4):1258-1262.
[18]Perlman, M. D. One-sided problems in multivariate analysis. Ann. Math. Statist.,1969,40(2): 549-567.
[19]Li K.C.,Several Relative distributions of Matrix Γ-distribution[J].J.of Math.,2008,28(5): 483-448
[20]Wortg C. S.On the use of differentials in statistics Linear[J].Math. Statist.1985,70:282-299.
[21]方开泰.实用多元分析[M].上海：华东师范大学出版社,1989.
[22]Haft L R. An identity for Wishart distribution with application[J]. Muhivar Analy,1979,9(4): 531-544.
[23]Kullback S. Information Theory and Statistics[M]. New York,Wiley,1959.
[24]Zhang Y.T., Bayesian inference in statistical analysis, Beijing,Science,1999.
[25]刘金山Wishart分布引论[M].北京：科学出版社,2005.
[26]李开灿,孟赵玲.χ2分布、t分布和F分布的一致渐进正态性[J].北京印刷学院学报.2004,12(3)：30—33.
[27]周如旗,陈文伟.基于EKLD的属性约简方法[J].计算机工程,2002,33(11)：62-63.
[28]白鹏.两个多元t分布之间的Kullback-Leibler距离[J].数学物理学报,2005,25(5)：694-709.
[29]张光远.关于多元t分布的一些讨论[J].新疆大学学报,1996,13(3)：33-38.
[30]Li K.C.,The moment for matrix Γ-distribution[J], Journal of Hubei Normal University, 2004,24(4):9-12.
[2]Konno, Y., The Exact moments of multivariate F and Beta distribution [J]. Japan. Statist. Soc,1980,18:123-130.
[3]Khatri, C.G.,A density free approach to the matrix variate beta distribution. Sankhya series, 1970,32(1):311-318.
[4]Mitra, S.K., A identity-free approach to matrix-variate Beta distribution[J]. Sankhya series, 1970,32(1):81-88.
[5]Bellman,R.Introduction to Matrix Analysis. [M]. New York, Springer,1970.
[6]Cramer,H.Mathematical methds of statistics[M].Princenton Univ.Press, Princenton,1946.
[7]Olkin,I.,Rubin,H.,Multivariate Beta distribution and independent property with the Wishart Distribution[J].Ann.Math. Statist.1964,35(1):261-269.
[8]Fang, B.Q., Dawid, A.P., Nonconjugate Bayesian regression on many variables[J].Statist, Planning and Inference.2003,103(1):245-261.
[9]Akimoto, Y. The uniform asymptotic normality of the Wishart Based on the K-L Infrmation [J]. J.Japan.Statist.soc.1994,24(1):73-81.
[10]Muirhead, R.J., Aspects of multivariate statistical theory[M]. Wikey, New York.1982.
[11]Apostol.T.,Calculus.Vol.1,Jlhn wiley & Sons, New York,1961.
[12]Robert G O,Shau S K. Updating Schemes,Correlation Structure,Blocking and Parameteriz for the Gibbs Samplerl [J]. J R Statist,1997,59:291-317.
[13]Reiss R D. Approximate Distributions of Order Statistics.[M]. New York,Springer,1980.
[14]张尧庭,方开泰.多元统计分析引论[M].北京：科学出版社,1982.
[15]李开灿.矩阵Γ分布的一致渐近正态性[J].数学学报,2006,49(2)：435-442.
[16]Whittaker J. Graphical Models in Applied Multivariate Statistics [M]. Wiley,Chichester,1990.
[17]Khatri,C. G. On the mutual independence of certain statistics[J]. Ann. Math. Statist.1959, 30(4):1258-1262.
[18]Perlman, M. D. One-sided problems in multivariate analysis. Ann. Math. Statist.,1969,40(2): 549-567.
[19]Li K.C.,Several Relative distributions of Matrix Γ-distribution[J].J.of Math.,2008,28(5): 483-448
[20]Wortg C. S.On the use of differentials in statistics Linear[J].Math. Statist.1985,70:282-299.
[21]方开泰.实用多元分析[M].上海：华东师范大学出版社,1989.
[22]Haft L R. An identity for Wishart distribution with application[J]. Muhivar Analy,1979,9(4): 531-544.
[23]Kullback S. Information Theory and Statistics[M]. New York,Wiley,1959.
[24]Zhang Y.T., Bayesian inference in statistical analysis, Beijing,Science,1999.
[25]刘金山Wishart分布引论[M].北京：科学出版社,2005.
[26]李开灿,孟赵玲.χ2分布、t分布和F分布的一致渐进正态性[J].北京印刷学院学报.2004,12(3)：30—33.
[27]周如旗,陈文伟.基于EKLD的属性约简方法[J].计算机工程,2002,33(11)：62-63.
[28]白鹏.两个多元t分布之间的Kullback-Leibler距离[J].数学物理学报,2005,25(5)：694-709.
[29]张光远.关于多元t分布的一些讨论[J].新疆大学学报,1996,13(3)：33-38.
[30]Li K.C.,The moment for matrix Γ-distribution[J], Journal of Hubei Normal University, 2004,24(4):9-12.
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