Shang Han Lin
Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, Australia.
Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, Australia.
J Appl Stat. 2020 Mar 3;48(4):583-604. doi: 10.1080/02664763.2020.1736527. eCollection 2021.
This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously estimate the bandwidths in the regression function and kernel error density by minimizing the Kullback-Leibler divergence. For estimating the regression function and error density, a series of simulation studies demonstrate that the functional partial linear model gives improved estimation and forecast accuracies compared with the functional principal component regression and functional nonparametric regression. Using a spectroscopy dataset, the functional partial linear model yields better forecast accuracy than some commonly used functional regression models. As a by-product of the Bayesian method, a pointwise prediction interval can be obtained, and marginal likelihood can be used to select the optimal semi-metric.
本研究考察了函数型部分线性模型的带宽和半度量的最优选择。我们提出的方法首先使用残差的核密度估计器来估计未知的误差密度，其中由参数和非参数分量组成的回归函数可以通过函数主成分估计器和函数 Nadayara-Watson 估计器来估计。回归函数和误差密度的估计精度关键取决于带宽和半度量的最优估计。利用贝叶斯方法通过最小化 Kullback-Leibler 散度来同时估计回归函数中的带宽和核误差密度。对于回归函数和误差密度的估计，一系列模拟研究表明，与函数主成分回归和函数非参数回归相比，函数型部分线性模型具有更高的估计和预测精度。使用一个光谱数据集，函数型部分线性模型比一些常用的函数回归模型具有更好的预测精度。作为贝叶斯方法的一个副产品，可以得到逐点预测区间，并且边际似然可用于选择最优半度量。
