Penalized Sparse Covariance Regression with High Dimensional Covariates
提出稀疏协方差回归方法，用惩罚技术从高维相似矩阵中筛选重要变量并估计系数，理论推导了误差界，并在中国股市数据上验证了有效性。
Covariance regression offers an effective way to model the large covariance\nmatrix with the auxiliary similarity matrices. In this work, we propose a\nsparse covariance regression (SCR) approach to handle the potentially\nhigh-dimensional predictors (i.e., similarity matrices). Specifically, we use\nthe penalization method to identify the informative predictors and estimate\ntheir associated coefficients simultaneously. We first investigate the Lasso\nestimator and subsequently consider the folded concave penalized estimation\nmethods (e.g., SCAD and MCP). However, the theoretical analysis of the existing\npenalization methods is primarily based on i.i.d. data, which is not directly\napplicable to our scenario. To address this difficulty, we establish the\nnon-asymptotic error bounds by exploiting the spectral properties of the\ncovariance matrix and similarity matrices. Then, we derive the estimation error\nbound for the Lasso estimator and establish the desirable oracle property of\nthe folded concave penalized estimator. Extensive simulation studies are\nconducted to corroborate our theoretical results. We also illustrate the\nusefulness of the proposed method by applying it to a Chinese stock market\ndataset.\n