Least squares estimation in the monotone single index model

Abstract

We study the monotone single index model where a real response variable $Y$ is linked to a $d$-dimensional covariate $X$ through the relationship $E[Y|X]=\Psi_{0}(\alpha^{T}_{0}X)$, 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 $\Psi_{0}({\alpha}^{T}_{0}\cdot)$ 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 $\alpha_{0}$ 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 $n^{1/3}$.