Asymptotic Distributions of Estimators of Eigenvalues and Eigenfunctions in Functional Data

Functional data analysis is a relatively new and rapidly growing area of statistics. This is partly due to technological advancements which have made it possible to generate new types of data that are in the form of curves. Because the data are functions, they lie in function spaces, which are of infinite dimension. To analyse functional data, one way, which is widely used, is to employ principal component analysis, allowing finite dimensional analysis of the problem. The authors gave stochastic expansions of estimators eigenvalues and eigenfunctions, providing not only a new understanding of the effects of truncating to a finite number of principal components, but also pointing to new methodology, such as simultaneous and individual bootstrap confidence statements for eigenvalues and eigenfunctions. The expansions explicitly include terms of sizes n?1/2, n?1, and a remainder of order n?3/2, where n denotes sample size. The terms of size n?1/2 are related to limit theory. Because for many situations, the exact statistical properties of the eigenvalues and eigenfunctions estimators are not directly obtainable, the way by which we can approximate their distributions is of interest in practice. In this paper, we discuss asymptotic results for eigenvalues and eigenfunctions. The work shows that eigenvalue spacings have only a second-order effect on properties of eigenvalue estimators, but a first-order effect on properties of eigenfunction estimators.