A Class of Linear Regression Parameter Estimators Constructed by Nonparametric Estimation

Given a $(p + 1)$-dimensional random vector $(X, Y)$ where f is the unknown density of X, the parameters of the multiple linear regression function α(x) = E(Y/X = x) = xβ may be estimated from a sample \(X₁, Y₁), ⋯, (Xₙ, Yₙ)\ by minimizing the functional ψ̂(β) = ∫(α̂ₙ(x) - xβ)²f̂ₙ(x) dx, where α̂ₙ and f̂ₙ may be any of a large class of nonparametric estimators of α and f. The strong consistency and asymptotic normality of the estimators so obtained are proved in this article under conditions on $(X, Y)$ that are less restrictive than those assumed by Faraldo Roca and Gonzalez Manteiga for $p = 1$. This class of estimators includes ordinary and generalized ridge regression estimators as special cases.