Quadratic shrinkage for large covariance matrices

This paper constructs a new estimator for large covariance matrices by drawing bridge between the classic (Stein (1975)) in finite samples and recent progress under large-dimensional asymptotics. The keeps eigenvectors of sample matrix applies shrinkage to inverse eigenvalues. corresponding formula is quadratic: it has two targets weighted quadratic functions concentration (that is, dimension divided size). first target dominates mid-level concentrations second one higher levels. extra degree freedom enables us outperform linear when optimal not linear, which general case. Both our are based on what we term “Stein shrinker”, local attraction operator that pulls eigenvalues towards their nearest neighbors, but whose force diminishes with distance (like gravitation). We prove no cubic or higher-order nonlinearities beat respect Frobenius loss Non-normality case where exceeds size accommodated. Monte Carlo simulations confirm state-of-the-art performance terms accuracy, speed, scalability.