Reviving pseudo-inverses: Asymptotic properties of large dimensional Moore–Penrose and ridge-type inverses with applications

Abstract

In this paper, we derive high-dimensional asymptotic properties of the Moore–Penrose inverse and, as a byproduct, of various ridge-type inverses of the sample covariance matrix. In particular, the analytical expressions of the asymptotic behavior of the weighted sample trace moments of generalized inverse matrices are deduced in terms of the partial exponential Bell polynomials, which can be easily computed in practice. The existent results for pseudo-inverses 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 provide universal methodology for construction of fully data-driven improved shrinkage estimators of the precision matrix, optimal portfolio weights and beyond. It is found that the Moore–Penrose inverse acts asymptotically as a certain regularizer of the true covariance matrix and it seems that its proper transformation (shrinkage) performs similar to or even outperforms the existing benchmarks in many applications, while keeping the computational time as minimal as possible.