On extended partially linear single-index models

Aiming to explore the relation between the response y and the stochastic explanatory vector variable X beyond the linear approximation, we consider the single-index model, which is a well-known approach in multidimensional cases. Specifically, we extend the partially linear single-index model to take the from y = beta(0)(T)X + phi(theta(0)(T)X) + epsilon, where epsilon is a random variable with E epsilon = 0 and var(epsilon)= sigma(2), unknown, beta(0) and theta(0) are unknown parametric vectors and phi(.) is an unknown real function. The model is also applicable to nonlinear time series analysis. In this paper, we propose a procedure to estimate the model and prove some related asymptotic results. Both simulated and real data are used to illustrate the results.