An alternating determination–optimization approach for an additive multi-index model

Sufficient dimension reduction techniques are to deal with curse of dimensionality when the underlying model is of a very general semiparametric multi-index structure and to estimate the central subspace spanned by the indices. However, the cost is that they can only identify the central subspace/central mean subspace and its dimension, rather than the indices themselves. In this paper, we investigate estimation for an additive multi-index model (AMM) that is of an additive structure with indices. The problem for AMM involves determining and estimating the nonparametric component functions and estimating the corresponding indices in the model. Different from the classical sufficient dimension reduction techniques in the estimation of the subspace and dimensionality determination, we propose a new penalized method to implement the estimation of component functions and of indices simultaneously. To this end, we suggest an alternating determination–optimization algorithm to alternatively fit best model and estimate the indices. Estimation consistency is provided. Simulation studies are carried out to examine the performance of the new method and a real data example is also analysed for illustration.