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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1751_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-type as well as in the limit distribution of degenerate <i>U</i>-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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月20日

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  • Bruno Ebner,
  • M. Dolores Jiménez-Gamero,
  • Bojana Milošević

摘要

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.