We consider variable selection for high-dimensional multivariate regression using penalized likelihoods when the number of outcomes and the number of covariates might be large. To account for within-subject correlation, we consider variable selection when a working precision matrix is used and when the precision matrix is jointly estimated using a two-stage procedure. We show that under suitable regularity conditions, penalized regression coefficient estimators are consistent for model selection for an arbitrary working precision matrix, and have the oracle properties and are efficient when the true precision matrix is used or when it is consistently estimated using sparse regression. We develop an efficient computation procedure for estimating regression coefficients using the coordinate descent algorithm in conjunction with sparse precision matrix estimation using the graphical LASSO (GLASSO) algorithm. We develop the Bayesian Information Criterion (BIC) for estimating the tuning parameter and show that BIC is consistent for model selection. We evaluate finite sample performance for the proposed method using simulation studies and illustrate its application using the type II diabetes gene expression pathway data.

中文翻译：

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高维多变量结果的变量选择

当结果的数量和协变量的数量可能很大时，我们考虑使用惩罚似然的高维多元回归的变量选择。为了考虑受试者内的相关性，我们在使用工作精度矩阵和使用两阶段程序联合估计精度矩阵时考虑变量选择。我们表明，在合适的正则条件下，惩罚回归系数估计器对于任意工作精度矩阵的模型选择是一致的，并且具有预言机属性，并且在使用真实精度矩阵或使用稀疏回归一致估计时是有效的。我们开发了一种有效的计算程序，用于使用坐标下降算法结合使用图形 LASSO (GLASSO) 算法的稀疏精度矩阵估计来估计回归系数。我们开发了贝叶斯信息准则 (BIC) 来估计调整参数，并表明 BIC 对于模型选择是一致的。我们使用模拟研究评估所提出方法的有限样本性能，并使用 II 型糖尿病基因表达途径数据说明其应用。