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|X]=Ψ₀(αT₀X), almost surely. Both the ridge function, Ψ₀, and the index parameter, α₀, are unknown and the ridge function is assumed to be monotone. Under some appropriate conditions, we show that the rate of convergence in the L₂-norm for the least squares estimator of the bundled function Ψ₀(αT₀·) is n1/3. A similar result is established for the isolated ridge function, and the index is shown to converge at least at the rate n1/3. Since the least squares estimator of the index is computationally intensive, we also consider alternative estimators of the index α₀ from earlier literature. Moreover, we show that if the rate of convergence of such an alternative estimator is at least n1/3, 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 n1/3.