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.

翻译：我们提出了一种针对高维设置下精度矩阵二次型函数的新型估计框架，特别适用于特征维度$p$超过样本量$n$的情形。传统的基于矩估计并经过偏差校正的方法在$p<n$（即$p/n \to c <1$）时仍能保持一致性。然而，一旦$p>n$，这些方法将完全失效，这凸显了由于秩缺失与高维复杂性所导致的两种情形之间的根本差异。我们的方法通过将谱矩表示与约束优化相结合，解决了上述问题，从而在温和的矩条件下实现了一致性估计。所提出的框架为一大类高维统计量的推断提供了统一方法。我们通过两个代表性示例说明其应用价值：投资组合优化中的最优夏普比率，以及回归分析中的多重相关系数。模拟研究表明，所提出的估计量能有效突破传统方法失效的$p>n$这一根本性障碍。