Semi-linear index model when the linear covariates and indices are independent

This chapter develops a flexible dimension-reduction model that incorporates both discrete and continuous covariates. Under this model, some covariates, Z, are related to the response variable, Y, through a linear relationship, while the remaining covariates, X, are related to Y through k indices which depend only on X’B and some unknown function g of X’B. To avoid the curse of dimensionality, k should be much smaller than p. This is often realistic as the key features of a high dimensional variable can often be extracted through a low-dimensional subspace. We develop a simple approach that separates the dimension reduction stage to estimate B from the remaining model components when the two covariates Z and X are independent. For instance, one can apply any suitable dimension reduction approach, such as the average derivative method, projection pursuit regression or sliced inverse regression, to get an initial estimator for B which is consistent at the √n rate, and then estimate the regression coefficient of Z and the link function g through a profile approach such as partial regression. All three estimates can be refined by iterating the procedure once. Such an approach is computationally simple and yields efficient estimates for both parameters at the √n rate. We provide both theoretical proofs and empirical evidence.