Equivalent Kernels of Local Polynomial Fitting in Varying Coefficient Models

• Activity Number: 346 - Recent Advances in Nonparametric Statistical Methods
• Type: Contributed
• Date/Time: Tuesday, July 31, 2018 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Chunyen Wu* and Li-Shan Huang
• Companies: Institute of Statistics, National Tsing Hua University, TAIWAN and Institute of Statistics, National Tsing Hua University, TAIWAN
• Keywords: Coefficient function; Kronecker product; Nonlinear interaction; Projection matrix

Abstract

The equivalent kernels for univariate local polynomial regression (Fan and Gijbels, (1996)) are well known; however, there are still no results for equivalent kernels in varying coefficient models. The aims of this paper are to give the analytic forms of those equivalent kernels and to discuss their moment properties in both finite and asymptotic cases. Our work shows that there is a particularly direct connection between the equivalent kernels of varying coefficient models and those of local polynomial regression. The equivalent kernels of varying coefficient models can be decomposed into two parts: the equivalent kernels of univariate local polynomial regression and linear combinations of covariates and conditional moments of covariates given the value of the smoothing variable, and the two parts are combined by the Kronecker product. These results give rise to a new projection-type estimate for varying coefficient models. We also derive the asymptotic properties for the new estimate and its asymptotic bias decreases as the order of polynomial fitting increases. Finally, some graphical results are given to illustrate weighting scheme of the equivalent kernels.