We introduce an estimation method of covariance matrices in a high-dimensional setting, i.e., when the dimension of the matrix, p, is larger than the sample size n. Specifically, we propose an orthogonally equivariant estimator. The eigenvectors of such estimator are the same as those of the sample covariance matrix. The eigenvalue estimates are obtained from an adjusted profile likelihood function derived by approximating the integral of the density function of the sample covariance matrix over its eigenvectors, which is a challenging problem in its own right. Exact solutions to the approximate likelihood equations are obtained and employed to construct estimates that involve a tuning parameter. Bootstrap and cross-validation based algorithms are proposed to choose this tuning parameter under various loss functions. Finally, comparisons with two well-known orthogonally equivariant estimators are given, which are based on Monte-Carlo risk estimates for simulated data and misclassification errors in real data analyses.

中文翻译：

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高维和小样本量的协方差矩阵的正交等变估计

我们在高维设置中引入协方差矩阵的估计方法，即当矩阵的维数 p 大于样本大小 n 时。具体来说，我们提出了一个正交等变估计器。这种估计器的特征向量与样本协方差矩阵的特征向量相同。特征值估计是从调整后的轮廓似然函数中获得的，该函数是通过近似样本协方差矩阵的密度函数对其特征向量的积分而得出的，这本身就是一个具有挑战性的问题。获得近似似然方程的精确解并将其用于构建涉及调整参数的估计。提出了基于引导和交叉验证的算法来在各种损失函数下选择这个调整参数。最后，