协方差矩阵的乘积及其迹的估计  

On the estimation for product of covariance matrices and its trace

作　　者：沈洁琼[1,2] 周杰 陈晨[1] SHEN Jie-Qiong;ZHOU Jie;CHEN Chen(School of Mathematics, Sichuan University, Chengdu 610064, China;School of Computer and Data Engineering, Ningbo Tech University, Ningbo 315100, China)

机构地区：[1]四川大学数学学院,成都610064 [2]浙大宁波理工学院计算机与数据工程学院,宁波315100

出　　处：《四川大学学报（自然科学版）》2020年第4期635-641,共7页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(61374027,11871357);四川省应用基础研究项目(2019YJ0122)。

摘　　要：对于两个不同总体的协方差矩阵Σ1和Σ2,估计其乘积Σ1Σ2及乘积的迹tr(Σ1Σ2)是统计推断问题的关键步骤.本文首先构造了Σ1Σ2的几个等价估计,并对任意的正整数m,n建立了Σm1Σn2和(Σ1Σ2)m的无偏估计.其次,利用Σ1Σ2的等价估计,本文证明tr(Σ1Σ2)的多个常用估计量是相等的.本文最后证明了两个常用的检验统计量(被用于检验两个协方差矩阵是否相等)是渐近等价的.In many statistical inference problems involving two populations respectively with covariance matricesΣ1 andΣ2,the productΣ1Σ2 and its trace tr(Σ1Σ2)need to be estimated.Firstly,we construct some equivalent estimators ofΣ1Σ2 and unbiased estimators ofΣm1Σn2 and(Σ1Σ2)m for any positive integers m and n.Secondly,by using the equivalent estimators ofΣ1Σ2,we show that some existing estimators of tr(Σ1Σ2)are equal.Finally,it is proved that two existing test statistics for testing the equality of two covariance matrices are asymptotically equivalent.

关 键 词：协方差矩阵 估计 统计推断 等价 

分 类 号：O212.4[理学—概率论与数理统计]