Regularized multivariate regression models with skew-t error distributions

• Regularized estimators of the regression parameters and the precision matrix for the multivariate linear regression with skew-t errors.
• Developing EM algorithm by taking advantage of the hierarchical representation of the multivariate skew-t distributions.
• Using simulation to assess the performance of the iterative algorithms for maximizing the penalized likelihood.
• Application to real data analysis.

We consider regularization of the parameters in multivariate linear regression models with the errors having a multivariate skew-t distribution. An iterative penalized likelihood procedure is proposed for constructing sparse estimators of both the regression coefficient and inverse scale matrices simultaneously. The sparsity is introduced through penalizing the negative log-likelihood by adding L1-penalties on the entries of the two matrices. Taking advantage of the hierarchical representation of skew-t distributions, and using the expectation conditional maximization (ECM) algorithm, we reduce the problem to penalized normal likelihood and develop a procedure to minimize the ensuing objective function. Using a simulation study the performance of the method is assessed, and the methodology is illustrated using a real data set with a 24-dimensional response vector.