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

TL;DR

This paper introduces a new class of dense, consistent, and model-free precision matrix estimators that unify these features within a nonasymptotic framework, revealing novel phenomena like double descent in precision estimation.

Contribution

It proposes a general, model-free estimator class for precision matrices that achieves density and consistency simultaneously, and uncovers the double descent phenomenon in this context.

Findings

Ridgeless regression exhibits double descent in precision matrix estimation.

Empirical study on S&P 500 index shows a doubly ascending Sharpe ratio pattern.

Theoretical analysis characterizes estimation error via complexity, signal, and bias.

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.