Efficient eigenvalue approximation in covariance operators via Rayleigh–Ritz with statistical applications

Bruno Ebner (), M. Dolores Jiménez-Gamero () and Bojana Milošević ()
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Bruno Ebner: Karlsruhe Institute of Technology (KIT)
M. Dolores Jiménez-Gamero: Universidad de Sevilla
Bojana Milošević: University of Belgrade

Statistical Papers, 2025, vol. 66, issue 6, No 12, 40 pages

Abstract: Abstract Finding the eigenvalues connected to the covariance operator of a centered Hilbert-space valued Gaussian process is genuinely considered a hard problem in several mathematical disciplines. In Statistics this problem arises for instance in the asymptotic null distribution of goodness-of-fit test statistics of weighted $$L^2$$ -type as well as in the limit distribution of degenerate U-statistics. For this problem we present the Rayleigh–Ritz method to approximate the eigenvalues. The usefulness of these approximations is shown by high lightening implications such as critical value approximation and theoretical comparison of test statistics by means of Bahadur efficiencies.

Keywords: Covariance operator; Eigenvalues; Rayleigh-Ritz method; Gaussian Processes; Statistics (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00362-025-01751-5

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