Estimation of a groupwise additive multiple-index model and its applications

In this paper, we propose a simple linear least squares framework to deal with estimation and selection for a groupwise additive multiple-index model, of which the partially linear single-index model is a special case, and in which each component function has a single-index structure. We show that, somewhat unexpectedly, all index vectors can be recovered through a single least squares coefficient vector. As a direct application, for partially linear single-index models we develop a new two-stage estimation procedure that is iterative-free and easily implemented. This estimation approach can also be applied to develop, for the semi-parametric model under study, a penalized least squares estimation and establish its asymptotic behavior in sparse and high-dimensional settings without any nonparametric treatment. A simulation study and a data analysis are presented.