Local Rank Inference for Varying Coefficient Models

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.

[1]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[2]  Thomas M. Stoker,et al.  Semiparametric Estimation of Index Coefficients , 1989 .

[3]  Jianqing Fan,et al.  Statistical Estimation in Varying-Coefficient Models , 1999 .

[4]  J. L. Hodges,et al.  The Efficiency of Some Nonparametric Competitors of the t-Test , 1956 .

[5]  Joseph W. McKean,et al.  Rank-Based Analysis of Linear Models Using R , 2005 .

[6]  Z. Ying,et al.  A simple resampling method by perturbing the minimand , 2001 .

[7]  D. Pollard,et al.  Asymptotics for minimisers of convex processes , 2011, 1107.3806.

[8]  Jianqing Fan,et al.  AN OVERVIEW ON NONPARAMETRIC AND SEMIPARAMETRIC TECHNIQUES FOR LONGITUDINAL DATA , 2006 .

[9]  Chin-Tsang Chiang,et al.  Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model With Longitudinal Data , 1998 .

[10]  D. Pollard.  Asymptotics for Least Absolute Deviation Regression Estimators , 1991, Econometric Theory.

[11]  D. H. Leung,et al.  Cross-validation in nonparametric regression with outliers , 2005 .

[12]  Kee-Hoon Kang,et al.  Bootstrapping nonparametric density estimators with empirically chosen bandwidths , 2001 .

[13]  Jianqing Fan,et al.  Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models , 2000 .

[14]  Mi-Ok Kim,et al.  Quantile regression with varying coefficients , 2007, 0708.0471.

[15]  Gerhard Tutz,et al.  On model diagnostics using varying coefficient models , 1999 .

[16]  Jianqing Fan,et al.  Statistical Methods with Varying Coefficient Models. , 2008, Statistics and its interface.

[17]  Li Ping Yang,et al.  Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data , 1998 .

[18]  T. Hettmansperger,et al.  Robust Nonparametric Statistical Methods , 1998 .

[19]  J. Rice,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves , 1998 .

[20]  Stanley R. Johnson,et al.  Varying Coefficient Models , 1984 .

[21]  Wenyang Zhang,et al.  Variable Bandwidth Selection in Varying-Coefficient Models , 2000 .

[22]  M. C. Jones,et al.  Local Linear Quantile Regression , 1998 .

[23]  Joseph W. McKean,et al.  Robust Analysis of Linear Models , 2004 .

[24]  O. Linton.  Local Regression Models , 2010 .

[25]  Jianqing Fan,et al.  Efficient Estimation and Inferences for Varying-Coefficient Models , 2000 .

[26]  H. A. David.  Early sample measures of variability , 1998 .