On partial sufficient dimension reduction with applications to partially linear multi-index models

Partial dimension reduction is a general method to seek informative convex combinations of predictors of primary interest, which includes dimension reduction as its special case when the predictors in the remaining part are constants. In this article, we propose a novel method to conduct partial dimension reduction estimation for predictors of primary interest without assuming that the remaining predictors are categorical. To this end, we first take the dichotomization step such that any existing approach for partial dimension reduction estimation can be employed. Then we take the expectation step to integrate over all the dichotomic predictors to identify the partial central subspace. As an example, we use the partially linear multi-index model to illustrate its applications for semiparametric modeling. Simulations and real data examples are given to illustrate our methodology.