Positive Definite $\ell_1$ Penalized Estimation of Large Covariance Matrices
By Lingzhou Xue, Shiqian Ma, Hui Zou. 2012
177 citations recorded by the source corpus.
The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To simultaneously achieve sparsity and positive definiteness, we develop a positive definite $\ell_1$-penalized covariance estimator for estimating sparse large covariance matrices. An efficient alternating direction method is derived to solve the challenging optimization problem and its convergence properties are established. Under weak regularity conditions, non-asymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications.
Subfields: Computational Mechanics, Statistics and Probability, Signal Processing
View Paper: https://www.semanticscholar.org/paper/11ce5b2317e9348fadb894c5f7bd08236becbcbe
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Citations in the corpus
- Inverse Moment Methods for Sufficient Forecasting Using High-Dimensional Predictors (2016, unknown): “The error covariance matrix Σu can be estimated by thresholding the sample covariance matrix of the estimated residual xt − B̂f̂t, denoted by Σ̂u = (σ̂uij)p×p, as in Cai and Liu (2011), Xue et al. (2012), Fan et al. (2013) and Fan et al. (2016).”
- Analysis of elliptical copula correlation factor model with Kendall's tau (2013, methodology): “The procedure is suggested by a referee and is inspired by [45].”
- Covariance Regression Analysis (2017, unknown): “Following the suggestion of Xue, Ma, and Zou (2012), we set = 10−5.”
- High‐dimensional covariance matrix estimation (2019, background): “In the face of this problem, Xue et al. (2012) proposed an alternating direction algorithm for solving min where (cid:6) ≽ ϵ I p means that (cid:6) (cid:0) ϵ I p is positive semi-de(cid:12)nite, and (cid:6) 1 denotes the sum of absolute values of (cid:6) .”
- Multitask Quantile Regression under the Transnormal Model (2016, methodology): “Thus, the rank-based positive-definite (cid:2) 1 penalization is as follows: Remark that ˆ (cid:2) s 1 will guarantee the critical positive definite-ness, and it can be efficiently solved by the alternating direction method of multiplier in Xue, Ma, and Zou (2012).”