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Theory for high-order bounds in functional principal components analysis. (English) Zbl 1153.62050
Summary: Functional data analysis, or FDA, is a relatively new and rapidly growing area of statistics. A substantial part of the interest in the field derives from new types of data that are generated through the application of new technologies. Statistical methodologies, such as linear regression, which are effectively finite-dimensional in conventional statistical settings, become infinite-dimensional in the context of functional data. As a result, the convergence rates of estimators based on functional data can be relatively slow, and so there is substantial interest in methods for dimension reduction, such as principal components analysis (PCA). However, although the statistical development of PCA for FDA has been underway for approximately two decades, relatively high-order theoretical arguments have been largely absent. This makes it difficult to assess the impact that, for example, eigenvalue spacings have on properties of eigenvalue estimators, or to develop a concise first-order limit theory for linear functional regression.
This paper shows how to overcome these hurdles. It develops rigorous arguments that underpin stochastic expansions of estimators of eigenvalues and eigenfunctions, and shows how to use them to answer statistical questions. The theory is based on arguments from operator theory, made more challenging by the requirement of statisticians that closeness of functions be measured in the \(L_{\infty }\), rather than \(L_{2}\), metric. The statistical implications of the properties we develop have been discussed elsewhere, but the theoretical arguments that lie behind them have not been presented before.

MSC:
62H25 Factor analysis and principal components; correspondence analysis
62A01 Foundations and philosophical topics in statistics
46N30 Applications of functional analysis in probability theory and statistics
Software:
fda (R)
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