Asymptotic Normality of a Combined Regression Estimator

• Published in: Journal of multivariate analysis Vol. 71; no. 2; pp. 191 - 240
• Main Authors: Fan, Yanqin, Ullah, Aman
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.11.1999; Elsevier
• Series: Journal of Multivariate Analysis
• Subjects: asymptotic distribution; asymptotic distribution convex combination kernel estimation model specification testing parametric estimation semiparametric estimation; convex combination; Distribution theory; Exact sciences and technology; kernel estimation; Linear inference, regression; Mathematics; model specification testing; Nonparametric inference; parametric estimation; Probability and statistics; Sciences and techniques of general use; semiparametric estimation; Statistics; asymptotic distribution; semiparametric estimation; convex combination; kernel estimation; model specification testing; parametric estimation; Statistical regression; Density estimation; Asymptotic normality; Asymptotic behavior; Estimator robustness; Distribution function; Convergence rate; Non parametric estimation; Probability distribution; Parametric test; Convergence; Kernel method
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1006/jmva.1999.1838

In this paper, we propose a combined regression estimator by using a parametric estimator and a nonparametric estimator of the regression function. The asymptotic distribution of this estimator is obtained for cases where the parametric regression model is correct, incorrect, and approximately correct. These distributional results imply that the combined estimator is superior to the kernel estimator in the sense that it can never do worse than the kernel estimator in terms of convergence rate and it has the same convergence rate as the parametric estimator in the case where the parametric model is correct. Unlike the parametric estimator, the combined estimator is robust to model misspecification. In addition, we also establish the asymptotic distribution of the estimator of the weight given to the parametric estimator in constructing the combined estimator. This can be used to construct consistent tests for the parametric regression model used to form the combined estimator.