Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density
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In the context of semi-functional partial linear regression model, we study the problem of error density estimation. The unknown error density is approximated by a mixture of Gaussian densities with means being the individual residuals, and variance a constant parameter. This mixture error density has a form of a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, is estimated by the ordinary least squares and functional Nadaraya–Watson estimators. The estimation accuracy of the ordinary least squares and functional Nadaraya–Watson estimators jointly depends on the same bandwidth parameter. A Bayesian approach is proposed to simultaneously estimate the bandwidths in the kernel-form error density and in the regression function. Under the kernel-form error density, we derive a kernel likelihood and posterior for the bandwidth parameters. For estimating the regression function and error density, a series of simulation studies show that the Bayesian approach yields better accuracy than the benchmark functional cross validation. Illustrated by a spectroscopy data set, we found that the Bayesian approach gives better point forecast accuracy of the regression function than the functional cross validation, and it is capable of producing prediction intervals nonparametrically.
KeywordsFunctional Nadaraya–Watson estimator Functional regression Gaussian kernel mixture Error density estimation Markov chain Monte Carlo
The author thanks Professors Rob Hyndman and Donald Poskitt for introducing him to functional data analysis, and Professors Xibin Zhang and Maxwell King for introducing him to Bayesian bandwidth estimation. The author would like to acknowledge Monash Sun Grid for its excellent parallel computing facility. Special thanks go to an associate editor and a referee for their insightful comments and suggestions, which led to a much improved manuscript.
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