Homogeneity Estimation in Multivariate Generalized Linear Models

Multivariate regression models have been extensively studied in the literature and applied in practice. It is not unusual that some predictors may make the same nonnull contributions to all the elements of the response vector, especially when the number of predictors is very large. For convenience, we call the set of such predictors as the homogeneity set. In this paper, we consider a sparse high-dimensional multivariate generalized linear models with coexisting homogeneity and heterogeneity sets of predictors, which is very important to facilitate the understanding of the effects of different types of predictors as well as improvement on the estimation efficiency. We propose a novel adaptive regularized method by which we can easily identify the homogeneity set of predictors and investigate the asymptotic properties of the parameter estimation. More importantly, the proposed method yields a smaller variance for parameter estimation compared to the ones that do not consider the existence of a homogeneity set of predictors. We also provide a computational algorithm and present its theoretical justification. In addition, we perform extensive simulation studies and present real data examples to demonstrate the proposed method.