Improved Estimation of Dissimilarities by Presmoothing Functional Data

Abstract

We examine the effect of presmoothing functional data on estimating the dissimilarities among objects in a dataset, with applications to cluster analysis and other distance methods, such as multidimensional scaling and statistical matching. We prove that a shrinkage method of smoothing results in a better estimator of the dissimilarities among a set of noisy curves. For a model with independent noise structure, the smoothed-data dissimilarity estimator dominates the observed-data estimator. For a dependent-error model—often applicable when the functional data are measured nearly continuously over some domain—an asymptotic domination result is given for the smoothed-data estimator. A simulation study indicates the magnitude of improvement provided by the shrinkage estimator and examines its behavior for heavy-tailed noise structure. The shrinkage estimator presented here combines Stein estimation and basis function-based linear smoothers in a novel manner. Statisticians increasingly analyze sizable sets of functional data, and the results in this article are a useful contribution to the theory of the effect of presmoothing on functional data analysis.