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 et al., 2014). While Zhang et al. (2014) 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.

中文翻译：

---

非参数回归模型中局部线性拟合的贝叶斯带宽估计

本文针对非参数回归模型中回归函数的局部线性估计量，提出了一种用于带宽估计的贝叶斯采样方法。在贝叶斯采样方法中，误差密度近似为高斯密度的位置混合密度，其中个体误差和方差是一个常数参数。这种混合密度具有误差核密度估计器的形式，称为核形式误差密度（参见 Zhang 等人，2014 年）。虽然张等人。(2014) 使用局部常数（也称为 Nadaraya-Watson）估计器来估计回归函数，我们将其扩展到局部线性估计器，从而产生更准确的估计。拟议的调查是由于缺乏数据驱动的方法来同时选择回归函数和内核形式误差密度的局部线性估计器中的带宽。将带宽视为参数，我们推导出近似（伪）似然和后验。仿真研究表明，在积分平方误差的标准下，所提出的带宽估计优于经验法则和交叉验证方法。提议的带宽估计方法通过涉及公司所有权集中度的非参数回归模型和涉及状态价格密度估计的模型进行验证。仿真研究表明，所提出的带宽估计在积分平方误差的标准下优于经验法则和交叉验证方法。提议的带宽估计方法通过涉及公司所有权集中度的非参数回归模型和涉及状态价格密度估计的模型进行验证。仿真研究表明，在积分平方误差的标准下，所提出的带宽估计优于经验法则和交叉验证方法。提议的带宽估计方法通过涉及公司所有权集中度的非参数回归模型和涉及状态价格密度估计的模型进行验证。