Adaptive banding covariance estimation for high‐dimensional multivariate longitudinal data

Abstract

AbstractModelling the covariance matrix of multiple responses in longitudinal data plays a vital role. It is more challenging than its univariate counterpart due to the presence of correlations among multiple responses. Using the modified Cholesky block decomposition, we impose an adaptive block‐banded structure on the Cholesky factor and sparsity on the innovation variance matrices via a novel convex hierarchical penalty and lasso penalty, respectively. The resulting adaptive block‐banding regularized estimator is fully data‐driven and has more flexibility than regular banding estimators. We develop an efficient alternative convex optimization algorithm using the Alternating Direction Method of Multipliers (ADMM) algorithm. The resulting estimators converge optimally in the Frobenius norm. We establish row‐specific support recovery for the precision matrix. Simulations and real data analysis show that the proposed estimator is better able to reveal banding sparsity patterns in data.