Inference for elliptical copula multivariate response regression models

Abstract

The estimation of the coefficient matrix in a multivariate response linear regression model is considered in situations where we can observe only strictly increasing transformations of the continuous responses and covariates. It is further assumed that the joint dependence between all the observed variables is characterized by an elliptical copula. Penalized estimators of the coefficient matrix are obtained in a high-dimensional setting by assuming that the coefficient matrix is either element-wise sparse or row-sparse, and by incorporating the precision matrix of the error, which is also assumed to be sparse. Estimation of the copula parameters is achieved by inversion of Kendall’s tau. It is shown that when the true coefficient matrix is row-sparse, the estimator obtained via a group penalty outperforms the one obtained via a simple element-wise penalty. Simulation studies are used to illustrate this fact and the advantage of incorporating the precision matrix of the error when the correlation among the components of the error vector is strong. Moreover, the use of the normal-score rank correlation estimator is revisited in the context of high-dimensional Gaussian copula models. It is shown that this estimator remains as the optimal estimator of the copula correlation matrix in this setting.