A Monte Carlo Investigation of Smoothing Methods for Error Density Estimation in Functional Data Analysis with an Illustrative Application to a Chemometric Data Set

Abstract

Functional data analysis is a eld in statistics that analyzes data which are dependent on time or space and from which inference can be conducted. Functional data analysis methods can estimate residuals from functional regression models that in turn require robust univariate density estimators for error density estimation. The accurate estimation of the error density from the residuals allows evaluation of the performance of functional regression estimation. Kernel density estimation using maximum likelihood cross-validation and Bayesian bandwidth selection techniques with a Gaussian kernel are reproduced and compared to least-squares cross-validation and plug-in bandwidth selection methods with an Epanechnikov kernel. For simulated data, Bayesian bandwidth selection methods for kernel density estimation are shown to give the minimum mean expected square error for estimating the error density, but are computationally ine cient and may not be adequately robust for real data. The (bounded) Epanechnikov kernel function is shown to give similar results as the Gaussian kernel function for error density estimation after functional regression. When the functional regression model is applied to a chemometric data set, the local least-squares cross-validation method, used to select the bandwidth for the functional regression estimator, is shown to give a signi cantly smaller mean square predicted error than that obtained with Bayesian methods.