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The estimation of large precision matrices is crucial in modern multivariate analysis. Traditional sparsity assumptions, while useful, often fall short of accurately capturing the dependencies among features. This article addresses this limitation by focusing on precision matrix estimation for multivariate data characterized by a flexible yet unknown group structure. We introduce a novel approach that begins with the detection of this unknown group structure, clustering features within the low-dimensional space defined by the leading eigenvectors of the sample covariance matrix. Following this, we employ group-wise multivariate response linear regressions, guided by the identified group memberships, to estimate the precision matrix. We rigorously establish the theoretical foundations of our proposed method for both group detection and precision matrix estimation. The superior numerical performance of our approach is demonstrated through comprehensive simulation experiments and a comparative analysis with established methods in the field. Additionally, we apply our method to a real breast cancer dataset, showcasing its practical utility and effectiveness. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.
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User-Friendly Covariance Estimation for Heavy-Tailed Distributions: A Survey and Recent Results
We offer a survey of recent results on covariance estimation for heavy- tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce element-wise and spectrum-wise truncation operators, as well as their M-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key observation is that the estimators needs to adapt to the sample size, dimensional- ity of the data and the noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate their practical use, we propose data-driven procedures that automatically calibrate the tuning parameters. We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.
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- Award ID(s):
- 1712956
- PAR ID:
- 10096975
- Date Published:
- Journal Name:
- Statistical science
- ISSN:
- 2168-8745
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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