Efficient Estimation and Inferences for Varying-Coefficient Models

Abstract This article deals with statistical inferences based on the varying-coefficient models proposed by Hastie and Tibshirani. Local polynomial regression techniques are used to estimate coefficient functions, and the asymptotic normality of the resulting estimators is established. The standard error formulas for estimated coefficients are derived and are empirically tested. A goodness-of-fit test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coefficient functions in a varying-coefficient model are constant or whether any covariates are statistically significant in the model. The null distribution of the test is estimated by a conditional bootstrap method. Our estimation techniques involve solving hundreds of local likelihood equations. To reduce the computational burden, a one-step Newton-Raphson estimator is proposed and implemented. The resulting one-step procedure is shown to save computational cost on an order of tens with no deterioration in performance, both asymptotically and empirically. Both simulated and real data examples are used to illustrate our proposed methodology.

[1]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[2]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[3]  P. Bickel.  One-Step Huber Estimates in the Linear Model , 1975 .

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

[5]  Hung Chen,et al.  Convergence Rates for Parametric Components in a Partly Linear Model , 1988 .

[6]  P. Speckman.  Kernel smoothing in partial linear models , 1988 .

[7]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[8]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[9]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[10]  Ruey S. Tsay,et al.  Functional-Coefficient Autoregressive Models , 1993 .

[11]  Jianqing Fan.  Local Linear Regression Smoothers and Their Minimax Efficiencies , 1993 .

[12]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[13]  M. Wand,et al.  Multivariate Locally Weighted Least Squares Regression , 1994 .

[14]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[15]  Theo Gasser,et al.  Finite-Sample Variance of Local Polynomials: Analysis and Solutions , 1996 .

[16]  Jianqing Fan,et al.  Generalized Partially Linear Single-Index Models , 1997 .

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

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

[19]  David Ruppert,et al.  Local Estimating Equations , 1998 .

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

[21]  H. Müller,et al.  Local Polynomial Modeling and Its Applications , 1998 .

[22]  Jianqing Fan,et al.  One-step local quasi-likelihood estimation , 1999 .

[23]  Jianqing Fan,et al.  Functional-Coefficient Regression Models for Nonlinear Time Series , 2000 .

[24]  Jian Zhang,et al.  Sieve likelihood ratio statistics and Wilks phenomenon , 1999 .

[25]  Jianqing Fan,et al.  Two‐step estimation of functional linear models with applications to longitudinal data , 1999 .

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

[27]  Zongwu Cai,et al.  Application of a local linear autoregressive model to BOD time series , 2000 .

[28]  A. Ullah,et al.  Nonparametric bootstrap tests for neglected nonlinearity in time series regression models , 2001 .