A Class of Linear Regression Parameter Estimators Constructed by Nonparametric Estimation

Abstract

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 $\alpha(x) = E(Y/X = x) = x\beta$ may be estimated from a sample $\{(X_1, Y_1), \cdots, (X_n, Y_n)\}$ by minimizing the functional $\hat{\psi}(\beta) = \int(\hat{\alpha}_n(x) - x\beta)^2\hat{f}_n(x) dx$, where $\hat{\alpha}_n$ and $\hat{f}_n$ may be any of a large class of nonparametric estimators of $\alpha$ 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.