Linear regression estimation in non-linear single index models

الأصل · EN

In this article, we consider the problem of estimating the index parameter α₀ in the single index model E[Y |X] = f₀(α₀ᵗ X) with f₀ the unknown ridge function defined on R, X a d-dimensional covariate and Y the response. We show that when X is Gaussian, then α₀ can be consistently estimated by regressing the observed responses Yᵢ, i = 1,..., n on the covariates X₁,..., Xₙ after centering and rescaling. The method works without any additional smoothness assumptions on f₀ and only requires that cov(f₀(α₀ᵗ X),α₀ᵗX) ≠ 0, which is always satisfied by monotone and non-constant functions f₀. We show that our estimator is asymptotically normal and give the expression with its asymptotic variance. The approach is illustrated through a simulation study.