Regularized estimation in sparse high-dimensional multivariate regression, with application to a DNA methylation study

In this article, we consider variable selection for correlated high dimensional DNA methylation markers as multivariate outcomes. A novel weighted square-root LASSO procedure is proposed to estimate the regression coefficient matrix. A key feature of this method is tuning-insensitivity, which greatly simplifies the computation by obviating cross validation for penalty parameter selection. A precision matrix obtained via the constrained ℓ1 minimization method is used to account for the within-subject correlation among multivariate outcomes. Oracle inequalities of the regularized estimators are derived. The performance of our proposed method is illustrated via extensive simulation studies. We apply our method to study the relation between smoking and high dimensional DNA methylation markers in the Normative Aging Study (NAS).