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Principal components analysis of regularly varying functions. (English) Zbl 1428.62258

Summary: The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not require this assumption. Instead, we assume that the observed functions are regularly varying. We derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components. We obtain a number of results on the convergence of moments and almost sure convergence. We apply the new theory to establish the consistency of the regression operator in a functional linear model.

MSC:

62H25 Factor analysis and principal components; correspondence analysis
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
62R10 Functional data analysis

Software:

fda (R)
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References:

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