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Eigenvalues approximation of integral covariance operators with applications to weighted $L^2$ statistics

Bruno Ebner, María Dolores Jiménez-Gamero, Bojana Milošević

TL;DR

This work tackles the problem of estimating eigenvalues of the covariance operator associated with centered Gaussian processes, a key step in the asymptotic theory for weighted L^2 goodness-of-fit statistics. It introduces the Rayleigh-Ritz projection onto problem-specific orthonormal bases to approximate the leading eigenvalues of the covariance kernel, and proves that these estimates converge to the true values as the basis size grows. The authors demonstrate the method across multiple supports (continuous, discrete, and multivariate) and link the eigenvalues to distributional approximations via the Pearson system and to local Bahadur efficiency calculations, enabling accurate critical-value approximations and efficient test comparisons. The approach provides a fast, deterministic alternative to Monte Carlo methods and is extensible to a broad class of kernels and data types, offering practical benefits for high-dimensional and complex-weighted statistics.

Abstract

Finding the eigenvalues connected to the covariance operator of a centred 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. 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.

Eigenvalues approximation of integral covariance operators with applications to weighted $L^2$ statistics

TL;DR

This work tackles the problem of estimating eigenvalues of the covariance operator associated with centered Gaussian processes, a key step in the asymptotic theory for weighted L^2 goodness-of-fit statistics. It introduces the Rayleigh-Ritz projection onto problem-specific orthonormal bases to approximate the leading eigenvalues of the covariance kernel, and proves that these estimates converge to the true values as the basis size grows. The authors demonstrate the method across multiple supports (continuous, discrete, and multivariate) and link the eigenvalues to distributional approximations via the Pearson system and to local Bahadur efficiency calculations, enabling accurate critical-value approximations and efficient test comparisons. The approach provides a fast, deterministic alternative to Monte Carlo methods and is extensible to a broad class of kernels and data types, offering practical benefits for high-dimensional and complex-weighted statistics.

Abstract

Finding the eigenvalues connected to the covariance operator of a centred 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 -type. 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.
Paper Structure (17 sections, 1 theorem, 33 equations, 8 tables)

This paper contains 17 sections, 1 theorem, 33 equations, 8 tables.

Table of Contents

  1. Introduction
  2. The eigenvalue problem
  3. The Rayleigh-Ritz method
  4. Applications
  5. Support $M=[0,1]$
  6. Support $M=[0,\infty)$
  7. Support $M=\mathbb{R}$
  8. Support $M = \mathbb{N}_0$
  9. Support $M=\mathbb{R}^d$
  10. Usefulness of eigenvalues estimation
  11. Distribution approximation
  12. Bahadur efficiency
  13. Comments and Outlook
  14. Alternative eigenvalue approximation approaches
  15. Monte Carlo approach
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let $\{\lambda_i\}_{i \geq 1}$ denote the eigenvalues of $K$, arranged in decreasing order and repeated according to their multiplicity. Let $\{\widehat{\lambda}_i\}_{i=1}^n$ represent the solutions (in $x$) of equation eq:det, also arranged in decreasing order and repeated according to their multip

Theorems & Definitions (7)

  • Theorem 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 5.1
  • Example 5.2