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

## CANCELLED

**Time: **
Wed 2023-04-12 15.15 - 16.00

**Location: **
Cramér room, Albano, house 1

**Participating: **
Nestor Parolya (Delft University of Technology)

### 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.