Sparsity-regularized skewness estimation for the multivariate skew normal and multivariate skew [formula omitted] distributions

Abstract

The multivariate skew normal (MSN) and multivariate skew t (MST) distributions have received considerable attention in the past two decades because of their appealing mathematical properties and their usefulness for modeling skewed data. We develop sparse regularization methodology for estimating the skewness parameters of these two distributions. This methodology facilitates skewness selection, i.e., the identification of those marginal indices of skewness, if any, that are equal to zero. Obstacles that render skewness selection infeasible for existing parameterizations of the two distributions are described, and a new parameterization that permits the circumvention of those obstacles is introduced. A penalized likelihood method for sparsity-based regularized skewness estimation using the new parameterization is proposed. Model selection consistency and the oracle property of the method are established. A simulation study demonstrates that the method is reasonably effective for skewness selection and, because of inclusion of a ridge penalty, is more effective at preventing the divergence of the shape parameter in small samples than the Q-penalty approach of Azzalini and Arellano-Valle (2013). The simulation study demonstrates further that our method may improve the estimation of all parameters of the MSN and MST distributions, not merely the skewnesses, when some of the skewnesses are zero.