In this paper, we derive high-dimensional asymptotic properties of the Moore-Penrose inverse and the ridge-type inverse of the sample covariance matrix. In particular, the analytical expressions of the weighted sample trace moments are deduced for both generalized inverse matrices and are present by using the partial exponential Bell polynomials which can easily be computed in practice. The existent results are extended in several directions: (i) First, the population covariance matrix is not assumed to be a multiple of the identity matrix; (ii) Second, the assumption of normality is not used in the derivation; (iii) Third, the asymptotic results are derived under the high-dimensional asymptotic regime. Our findings are used to construct improved shrinkage estimators of the precision matrix, which asymptotically minimize the quadratic loss with probability one. Finally, the finite sample properties of the derived theoretical results are investigated via an extensive simulation study.

翻译：本文推导了大维样本协方差矩阵的Moore-Penrose逆与岭型逆的高维渐近性质。具体而言，我们通过部分指数型贝尔多项式（该多项式在实践中易于计算）导出了两类广义逆矩阵的加权样本迹矩解析表达式。现有结果在以下方向得到扩展：(i) 不再假设总体协方差矩阵为单位阵的倍数；(ii) 推导过程未使用正态性假设；(iii) 渐近结果在高维渐近框架下导出。利用这些发现，我们构建了精度矩阵的改进型收缩估计量，该估计量能以概率1渐近最小化二次损失。最后，通过大规模模拟研究验证了理论结果的有限样本性质。