基于均值未知的高维协方差矩阵的估计

Estimation of High Dimensional Covariance Matrix Based on Unknown Mean

给出了一种基于均值未知情形下,高维协方差矩阵估计的新算法。即当矩阵的维数p大于样本容量n时,根据随机矩阵理论,通过样本协方差矩阵特征值的边缘密度函数和总体特征值的对数似然函数,得到目标矩阵特征值的估计量。基于收缩估计的思想,对目标矩阵特征值和样本协方差矩阵特征值进行收缩估计,通过特征值的估计得到高维协方差矩阵的一个新的估计量。数值模拟表明,对于多元正态的总体,高维协方差矩阵的新估计量较样本协方差矩阵的精度更好。

A new algorithm for estimating high dimensional covariance matrix based on unknown mean is presented.That is,when the dimension of the matrix,p,is larger than the sample size n,according to the random matrix theory,the estimators of the eigenvalues of the objective matrix are obtained through the marginal density function of the eigenvalues of the sample covariance matrix and the logarithmic likelihood function of the population eigenvalues.Based on the idea of shrinkage estimation,the eigenvalues of target matrix and sample covariance matrix are estimated,and a new estimator of the high-dimensional covariance matrix is obtained by estimating the eigenvalues.Numerical simulation shows that the new estimator of high-dimensional covariance matrix is more accurate than the sample covariance matrix for multivariate normal population.