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

Bruno Ebner; María Dolores Jiménez-Gamero; Bojana Milošević

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arXiv · 2408.08064

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

Bruno Ebner·María Dolores Jiménez-Gamero·Bojana Milošević

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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.

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Bruno Ebner, María Dolores Jiménez-Gamero, Bojana Milošević. 2024-08-15. Eigenvalues approximation of integral covariance operators with applications to weighted $L^2$ statistics. https://arxiv.org/abs/2408.08064

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