Bayesian bandwidth estimation and semi-metric selection for a functional partial linear model with unknown error density

ABSTRACT

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.

Keywords:

• Functional Nadaraya-Watson estimator
• scalar-on-function regression
• Gaussian kernel mixture
• Markov chain Monte Carlo
• error-density estimation
• spectroscopy

2010 Mathematics Subject Classifications:

• 97K80
• 62F15

Acknowledgments

The author would like to thank a reviewer for insightful comments and suggestions, which substantially improve the early version of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Han Lin Shang  http://orcid.org/0000-0003-1769-6430

Additional information

Funding

The author acknowledges a faculty research grant from the College of Business and Economics at the Australian National University (R62860 I704).