Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation

By Tommaso Cai, Zhao Ren, Harrison H. Zhou. 2016

243 citations recorded by the source corpus.

This is an expository paper that reviews recent developments on optimal estimation of structured high-dimensional covariance and precision matrices. Minimax rates of convergence for estimating several classes of structured covariance and precision matrices, including bandable, Toeplitz, and sparse covariance matrices as well as sparse precision matrices, are given under the spectral norm loss. Data-driven adaptive procedures for estimating various classes of matrices are presented. Some key technical tools including large deviation results and minimax lower bound arguments that are used in the theoretical analyses are discussed. In addition, estimation under other losses and a few related problems such as Gaussian graphical models, sparse principal component analysis, and hypothesis testing on the covariance structure are considered. Some open problems on estimating high-dimensional covariance and precision matrices and their functionals are also discussed.

Subfields: Statistics and Probability, Signal Processing

View Paper: https://www.semanticscholar.org/paper/c3e6c03109dd437ef2e3f47bcb32a40b76cd1ace

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

1. D. Xiu
2. Jelena Bradic
3. Zhao Ren
4. Yinchu Zhu
5. Yacine Aït-Sahalia
6. Chao Gao
7. Tommaso Cai
8. Mengjie Chen

In Signal Processing

1. D. Xiu
2. Yacine Aït-Sahalia
3. Zhao Ren
4. Chao Gao
5. Mengjie Chen
6. Yinchu Zhu
7. Tommaso Cai
8. Jelena Bradic

Citations in the corpus

• Incorporating Global Industrial Classification Standard Into Portfolio Allocation: A Simple Factor-Based Large Covariance Matrix Estimator With High-Frequency Data (2016, unknown): “Alternatively, the thresholding approach is also very popular, including Bickel and Levina (2008a, 2008b) and Cai and Liu (2011), etc. Fan and Liu (2013) and Zhou, Cai, and Ren (2014) presented a very comprehensive review of this literature.”
• Robust covariance and scatter matrix estimation under Huber’s contamination model (2015, background): “For a recent comprehensive review on this topic, see [9].”
• Breaking the curse of dimensionality in regression (2017, background): “These assumptions bear nice theoretical property: parameters of interest are well defined and estimable (see for example the minimax rates of estimation Cai and Guo (2016); Cai et al. (2016); Yuan and Zhou (2016)).”
• Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “See Cai et al. (2016) for a survey of the recent results on optimal and adaptive estimation, and for a discussion of some key technical tools used in the theoretical analyses. www.annualreviews.org • Hypothesis Testing for Covariance Structures 3 Parallel to estimation, there have been significant…”
• Robust Covariance Matrix Estimation via Matrix Depth (2015, background): “For a recent comprehensive review on this topic, see [9].”
• Using Principal Component Analysis to Estimate a High Dimensional Factor Model with High-Frequency Data (2017, methodology): “Alternative methods rely on various forms of thresholding (Bickel and Levina (2008a), Bickel and Levina (2008b), Cai and Liu (2011), Fryzlewicz (2013), and Zhou et al. (2014)) whereas the estimator in Fan et al. (2013) is designed for latent factor models.”