Least squares estimation in the monotone single index model

We study the monotone single index model where a real response variable Y is linked to a d-dimensional covariate X through the relationship E[Y vertical bar X] = Psi(0)(alpha X-T(0)), almost surely. Both the ridge function, Psi(0), and the index parameter, alpha(0), are unknown and the ridge function is assumed to be monotone. Under some appropriate conditions, we show that the rate of convergence in the L-2-norm for the least squares estimator of the bundled function Psi(0)(alpha(T)(0).) is n(1/3). A similar result is established for the isolated ridge function, and the index is shown to converge at least at the rate n(1/3). Since the least squares estimator of the index is computationally intensive, we also consider alternative estimators of the index oto from earlier literature. Moreover, we show that if the rate of convergence of such an alternative estimator is at least n(1/3), then the corresponding least-squares type estimators (obtained via a "plug-in" approach) of both the bundled and isolated ridge functions still converge at the rate n(1/3).