Nestor Parolya: Resurrecting pseudoinverse: Asymptotic properties of large Moore–Penrose inverse with applications

Abstract

We derive high-dimensional asymptotic properties of the Moore–Penrose inverse of the sample covariance matrix, i.e., the case when number of variables p is larger than the sample size n. We prove the convergence results related to the traces of weighted moments of the Moore–Penrose inverse matrix, which involve both its eigenvalues and eigenvectors. We extend previous findings in several directions: (i) first, the population covariance matrix is not assumed to be a multiple of the identity; (ii) second, the assumptions of normality is not used in the derivation, only existence of the 4th moments is required; (iii) third, the asymptotic properties of the weighted moments are derived under the high-dimensional asymptotic regime: p and n tend to infinity such that p/n tend to a constant greater than 1. Our findings allow the construction of the optimal linear shrinkage estimators for large precision matrix, beta-vector in the high-dimensional linear models and global minimum variance portfolio.