Model-Estimation-Free, Dense, and High Dimensional Consistent Precision Matrix Estimators

Abstract

Precision matrix estimation is a cornerstone concept in statistics, economics, and finance. Despite advances in recent years, estimation methods that are simultaneously (i) dense, (ii) consistent, and (iii) model-free are lacking. While each of these targets can be met separately, achieving them together is challenging.We address this gap by introducing a general class of estimators that unifies these features within a nonasymptotic framework, allowing for explicit characterization of the computational complexity, signal-to-noise ratio trade-off. Our analysis identifies three fundamental random quantities, complexity, signal magnitude, and method bias that jointly determine estimation error. A particularly striking result is that ridgeless regression, a tuning-free special case within our class, exhibits the double descent phenomenon. This establishes the first formal precision matrix analogue to the well-known double descent behavior in linear regression. Our theoretical analysis is supported by a thorough empirical study of the S\&P 500 index, where we observe a doubly ascending Sharpe ratio pattern, which complements the double descent phenomenon.