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June, 1987 A Class of Linear Regression Parameter Estimators Constructed by Nonparametric Estimation
J. A. Cristobal Cristobal, P. Faraldo Roca, W. Gonzalez Manteiga
Ann. Statist. 15(2): 603-609 (June, 1987). DOI: 10.1214/aos/1176350363


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.


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J. A. Cristobal Cristobal. P. Faraldo Roca. W. Gonzalez Manteiga. "A Class of Linear Regression Parameter Estimators Constructed by Nonparametric Estimation." Ann. Statist. 15 (2) 603 - 609, June, 1987.


Published: June, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0631.62041
MathSciNet: MR888428
Digital Object Identifier: 10.1214/aos/1176350363

Primary: 62J05
Secondary: 62G05

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 2 • June, 1987
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