High-Dimensional Precision Matrix Quadratic Forms: Estimation Framework for $p > n$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909983406292992 |
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| author | Hong, Shizhe Li, Weiming Pan, Guangming |
| author_facet | Hong, Shizhe Li, Weiming Pan, Guangming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03815 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-Dimensional Precision Matrix Quadratic Forms: Estimation Framework for $p > n$ Hong, Shizhe Li, Weiming Pan, Guangming Methodology 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. |
| title | High-Dimensional Precision Matrix Quadratic Forms: Estimation Framework for $p > n$ |
| topic | Methodology |
| url | https://arxiv.org/abs/2601.03815 |