Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss
Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss
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DOI：
10.1016/j.jmva.2009.05.002
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发表时间：
2009-11-01
影响因子：
1.6
通讯作者：
Konno, Yoshihiko
中科院分区：
文献类型：
--
作者：
Konno, Yoshihiko
研究了多元实正态分布和复正态分布中，当变量维数大于样本数时，大协方差矩阵的估计问题。分别针对实数和复数情况，给出了奇异Wishart矩阵的Stein-Haff恒等式和特征结构演算。利用这些技术，分别得到了在不变二次损失函数下，实情况和复情况下，总体协方差矩阵的某类估计量的无偏风险估计。在无偏风险估计的基础上，提出了与Haff [L.R.]估计量对应的收缩估计量李海峰，多变量正态协方差矩阵的经验贝叶斯估计，安。Statist. 8(1980) 586-697]表明，在变量维数大于样本数的情况下，对于实多元和复多元正态分布，在不变二次损失函数下，经验协方差矩阵的最佳标量乘法得到了改进。(C) 2009爱思唯尔公司版权所有。
The problem of estimating large covariance matrices of multivariate real normal and complex normal distributions is considered when the dimension of the variables is larger than the number of samples. The Stein-Haff identities and calculus on eigenstructure for singular Wishart matrices are developed for real and complex cases, respectively. By using these techniques, the unbiased risk estimates for certain classes of estimators for the population covariance matrices under invariant quadratic loss functions are obtained for real and complex cases, respectively. Based on the unbiased risk estimates, shrinkage estimators which are counterparts of the estimators due to Haff [L.R. Haff, Empirical Bayes estimation of the multivariate normal covariance matrix, Ann. Statist. 8 (1980) 586-697] are shown to improve upon the best scalar multiple of the empirical covariance matrix under the invariant quadratic loss functions for both real and complex multivariate normal distributions in the situation where the dimension of the variables is larger than the number of samples. (C) 2009 Elsevier Inc. All rights reserved.