Estimation of Semiparametric Multi-Index Models Using Deep Neural Networks

In this paper, we consider estimation and inference for both the multi-index parameters and the link function involved in a class of semiparametric multi-index models via deep neural networks (DNNs). We contribute to the design of DNN by i) providing more transparency for practical implementation, ii) defining different types of sparsity, iii) showing the differentiability, iv) pointing out the set of effective parameters, and v) offering a new variant of rectified linear activation function (ReLU), etc. Asymptotic properties for the joint estimates of both the index parameters and the link functions are established, and a feasible procedure for the purpose of inference is also proposed. We conduct extensive numerical studies to examine the finite-sample performance of the estimation methods, and we also evaluate the empirical relevance and applicability of the proposed models and estimation methods to real data.