On efficient dimension reduction with respect to the interaction between two response variables

In this paper, we propose the novel theory and methodologies for dimension reduction with respect to the interaction between two response variables, which is a new research problem that has wide applications in missing data analysis, causal inference, graphical models, etc. We formulate the parameters of interest to be the locally and the globally efficient dimension reduction subspaces, and justify the generality of the corresponding low‐dimensional assumption. We then construct estimating equations that characterize these parameters, using which we develop a generic family of consistent, model‐free and easily implementable dimension reduction methods called the dual inverse regression methods. We also build the theory regarding the existence of the globally efficient dimension reduction subspace, and provide a handy way to check this in practice. The proposed work differs fundamentally from the literature of sufficient dimension reduction in terms of the research interest, the assumption adopted, the estimation methods and the corresponding applications, and it potentially creates a new paradigm of dimension reduction research. Its usefulness is illustrated by simulation studies and a real data example at the end.