[1]朱道元,赵桂芹.Bartlett 分解定理的简便证明[J].东南大学学报(自然科学版),2001,31(5):125-127.[doi:10.3969/j.issn.1001-0505.2001.05.027]
 Zhu Daoyuan,Zhao Guiqin.A Concise Proof of the Bartlett Decomposition Theorem[J].Journal of Southeast University (Natural Science Edition),2001,31(5):125-127.[doi:10.3969/j.issn.1001-0505.2001.05.027]
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Bartlett 分解定理的简便证明()
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《东南大学学报(自然科学版)》[ISSN:1001-0505/CN:32-1178/N]

卷:
31
期数:
2001年第5期
页码:
125-127
栏目:
数学、物理学、力学
出版日期:
2001-09-20

文章信息/Info

Title:
A Concise Proof of the Bartlett Decomposition Theorem
作者:
朱道元 赵桂芹
东南大学应用数学系, 南京 210096
Author(s):
Zhu Daoyuan Zhao Guiqin
Department of Applied Mathematics, Southeast University, Nanjing 210096, China)
关键词:
Bartlett分解定理 特征函数 Wishart 分布
Keywords:
Bartlett decomposition theorem characteristic function Whishart distribution
分类号:
O212.1
DOI:
10.3969/j.issn.1001-0505.2001.05.027
摘要:
对服从Wishart 分布的随机矩阵W~Wp(n,Ι)已有著名的Bartlett分解定理,结果非常完美,但证明过程既繁又长,本文用特征函数方法证明2个服从n-i+1维标准正态分布、且相互独立的随机向量的内积应同分布于一个服从χn-i+1分布的随机变量与一个与其独立且服从N(0,1)分布的随机变量的乘积.从而简单而直观地证明该定理,虽结论稍减弱为W=dT′T,但并不影响其在大多数场合的应用.
Abstract:
For a random matrix that represents the Whishart distribution, there exists the well-known Bartlett decomposition theorem. The result is perfect, but the proof is tedious. This article uses the characteristic function to prove the following result: the inner product of two independent (n-i+1)-dimensional random vectors subject to standard normal distribution has the same distribution of the product of two independent random variables in which one is subject to χn-i+1 distribution and the other to N(0,1) distribution. Thus the theorem can be proved briefly and directly. Though the conclusion is reduced to W=dT′T, it does not affect the application of Bartlett decomposition theorem in most cases.

参考文献/References:

[1] Anderson T W.An introduction to multivariate statistical analysis.2nd Edition.New York:Wiley,1984.244~248
[2] Murihead R J.Aspects of multivariate statistical theory.New York:Wiley,1982.50~86
[3] Rao C R.Linear statistical inference and its applications.New York:Wiley,1973.533~537
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备注/Memo

备注/Memo:
作者简介:朱道元,男,1947年生,教授.
更新日期/Last Update: 2001-09-20