Multivariate bandwidth selection for local linear regression

The existence and properties of optimal bandwidths for multivariate local linear regression are established, using either a scalar bandwidth for all regressors or a diagonal bandwidth vector that has a different bandwidth for each regressor. Both involve functionals of the derivatives of the unknown multivariate regression function. Estimating these functionals is difficult primarily because they contain multivariate derivatives. In this paper, an estimator of the multivariate second derivative is obtained via local cubic regression with most cross-terms left out. This estimator has the optimal rate of convergence but is simpler and uses much less computing time than the full local estimator. Using this as a pilot estimator, we obtain plug-in formulae for the optimal bandwidth, both scalar and diagonal, for multivariate local linear regression. As a simpler alternative, we also provide rule-of-thumb bandwidth selectors. All these bandwidths have satisfactory performance in our simulation study.

[1]  Ming-Yen Cheng,et al.  A bandwidth selector for local linear density estimators , 1997 .

[2]  Wolfgang Karl Härdle,et al.  Nonparametric Vector Autoregression , 1998 .

[3]  M. C. Jones,et al.  Comparison of Smoothing Parameterizations in Bivariate Kernel Density Estimation , 1993 .

[4]  M. C. Jones,et al.  A Brief Survey of Bandwidth Selection for Density Estimation , 1996 .

[5]  D. W. Scott,et al.  Cross-Validation of Multivariate Densities , 1994 .

[6]  Nonparametric estimation of generalized impulse response function , 2000 .

[7]  J. Marron,et al.  Progress in data-based bandwidth selection for kernel density estimation , 1996 .

[8]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[9]  M. Wand,et al.  Multivariate plug-in bandwidth selection , 1994 .

[10]  M. Wand,et al.  An Effective Bandwidth Selector for Local Least Squares Regression , 1995 .

[11]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[12]  Wolfgang Härdle,et al.  Web Quantlets for Time Series Analysis , 2001 .

[13]  D. Ruppert.  Empirical-Bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation , 1997 .

[14]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[15]  A Bandwidth Selector for Bivariate Kernel Regression , 1995 .

[16]  Jianqing Fan,et al.  Adaptive Order Polynomial Fitting: Bandwidth Robustification and Bias Reduction , 1995 .

[17]  Lijian Yang,et al.  Nonparametric Lag Selection for Time Series , 2000 .

[18]  James Stephen Marron,et al.  Fast and simple scatterplot smoothing , 1995 .

[19]  H. Luetkepohl.  The Handbook of Matrices , 1996 .

[20]  Jörg Polzehl,et al.  Bias corrected bootstrap bandwidth selection , 1997 .

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