By allowing the regression coefficients to change with certain covariates, the class of varying coefficient models offers a flexible approach to modeling nonlinearity and interactions between covariates. This article proposes a novel estimation procedure for the varying coefficient models based on local ranks. The new procedure provides a highly efficient and robust alternative to the local linear least squares method, and can be conveniently implemented using existing R software package. Theoretical analysis and numerical simulations both reveal that the gain of the local rank estimator over the local linear least squares estimator, measured by the asymptotic mean squared error or the asymptotic mean integrated squared error, can be substantial. In the normal error case, the asymptotic relative efficiency for estimating both the coefficient functions and the derivative of the coefficient functions is above 96%; even in the worst case scenarios, the asymptotic relative efficiency has a lower bound 88.96% for estimating the coefficient functions, and a lower bound 89.91% for estimating their derivatives. The new estimator may achieve the nonparametric convergence rate even when the local linear least squares method fails due to infinite random error variance. We establish the large sample theory of the proposed procedure by utilizing results from generalized U-statistics, whose kernel function may depend on the sample size. We also extend a resampling approach, which perturbs the objective function repeatedly, to the generalized U-statistics setting, and demonstrate that it can accurately estimate the asymptotic covariance matrix.
Bibliographical noteFunding Information:
Lan Wang is Associate Professor, School of Statistics, University of Minnesota, Minneapolis, MN 55455 (E-mail: email@example.com). Bo Kai is Assistant Professor, Department of Mathematics, College of Charleston, Charleston, SC 29424 (E-mail: firstname.lastname@example.org). Runze Li is Professor, Department of Statistics and The Methodology Center, The Pennsylvania State University, University Park, PA 16802 (E-mail: email@example.com). Wang’s research is supported by National Science Foundation grant DMS-0706842. Kai’s research is supported by National Science Foundation grants DMS 0348869 as a research assistant. Li’s research is supported by NIDA, NIH grants R21 DA024260 and P50 DA10075. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA or the NIH. The authors thank an associate editor and two anonymous referees for their insightful and constructive comments.
- Asymptotic relative efficiency
- Local linear regression
- Local rank
- Varying coefficient model