Linear regression estimation in non-linear single index models

In this article, we consider the problem of estimating the index parameter ${\alpha }_0$ in the single index model $E[Y\mathrm{\mid }X]=f_0({\alpha }_0^{T}X)$ with $f_0$ the unknown ridge function defined on $\mathbb{R}$, $X$ a d-dimensional covariate and $Y$ the response. We show that when $X$ is Gaussian, then ${\alpha }_0$ can be consistently estimated by regressing the observed responses $Y_i$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},n$ on the covariates $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ after centering and rescaling. The method works without any additional smoothness assumptions on $f_0$ and only requires that $cov(f_0({\alpha }_0^{T}X),{\alpha }_0^{T}X)\mathrm{\ne }0$, which is always satisfied by monotone and non-constant functions $f_0$. We show that our estimator is asymptotically normal and give the expression with its asymptotic variance. The approach is illustrated through a simulation study.