Abstract: Multivariate regression model is a natural generalization of the classical univariate regression model for fitting multiple responses. In this paper, we propose a high-dimensional multivariate conditional regression model for constructing sparse estimates of the multivariate regression coefficient matrix that accounts for the dependency structure among the multiple responses. The proposed method decomposes the multivariate regression problem into a series of penalized conditional log-likelihood of each response conditional on the covariates and other responses. It allows simultaneous estimation of the sparse regression coefficient matrix and the sparse inverse covariance matrix. The asymptotic selection consistency and normality are established for the diverging dimension of the covariates and number of responses. The effectiveness of the proposed method is demonstrated in a variety of simulated examples as well as an application to the Glioblastoma multiforme cancer data.

Key words and phrases: Covariance selection, Gaussian graphical model, large p small n, multivariate regression, regularization.