High-Dimensional Precision Matrix Quadratic Forms: Estimation Framework for $p > n$

TL;DR

This paper introduces a new estimation framework for quadratic functionals of high-dimensional precision matrices that remains consistent even when the feature dimension exceeds the sample size, addressing a key challenge in high-dimensional statistics.

Contribution

The paper develops a spectral-moment and constrained optimization-based approach for consistent estimation of quadratic functionals in high-dimensional regimes where traditional methods fail.

Findings

Effective estimation of quadratic functionals when p > n

Simulation results show superiority over conventional methods

Framework applicable to portfolio optimization and regression analysis

Abstract

We propose a novel estimation framework for quadratic functionals of precision matrices in high-dimensional settings, particularly in regimes where the feature dimension $p$ exceeds the sample size $n$. Traditional moment-based estimators with bias correction remain consistent when $p<n$ (i.e., $p/n \to c <1$). However, they break down entirely once $p>n$, highlighting a fundamental distinction between the two regimes due to rank deficiency and high-dimensional complexity. Our approach resolves these issues by combining a spectral-moment representation with constrained optimization, resulting in consistent estimation under mild moment conditions. The proposed framework provides a unified approach for inference on a broad class of high-dimensional statistical measures. We illustrate its utility through two representative examples: the optimal Sharpe ratio in portfolio optimization and the multiple correlation coefficient in regression analysis. Simulation studies demonstrate that the proposed estimator effectively overcomes the fundamental $p>n$ barrier where conventional methods fail.