On extended partially linear single-index models

Abstract

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 form y = β0 TX + φ(θ0 TX) + ε, where ε is a random variable with Eε = 0 and var(ε) = σ2, unknown, β0 and θ0 are unknown parametric vectors and φ(.) 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. © 1999 Biometrika Trust.