Bayesian bandwidth estimation for local linear fitting in nonparametric regression models

Abstract

Abstract This paper presents a Bayesian sampling approach to bandwidth estimation for the local linear estimator of the regression function in a nonparametric regression model. In the Bayesian sampling approach, the error density is approximated by a location-mixture density of Gaussian densities with means the individual errors and variance a constant parameter. This mixture density has the form of a kernel density estimator of errors and is referred to as the kernel-form error density (c.f. Zhang, X., M. L. King, and H. L. Shang. 2014. “A Sampling Algorithm for Bandwidth Estimation in a Nonparametric Regression Model with a Flexible Error Density.” Computational Statistics & Data Analysis 78: 218–34.). While (Zhang, X., M. L. King, and H. L. Shang. 2014. “A Sampling Algorithm for Bandwidth Estimation in a Nonparametric Regression Model with a Flexible Error Density.” Computational Statistics & Data Analysis 78: 218–34) use the local constant (also known as the Nadaraya-Watson) estimator to estimate the regression function, we extend this to the local linear estimator, which produces more accurate estimation. The proposed investigation is motivated by the lack of data-driven methods for simultaneously choosing bandwidths in the local linear estimator of the regression function and kernel-form error density. Treating bandwidths as parameters, we derive an approximate (pseudo) likelihood and a posterior. A simulation study shows that the proposed bandwidth estimation outperforms the rule-of-thumb and cross-validation methods under the criterion of integrated squared errors. The proposed bandwidth estimation method is validated through a nonparametric regression model involving firm ownership concentration, and a model involving state-price density estimation.