Eigenvalues approximation of integral covariance operators with applications to weighted L² statistics

Abstract

We review methods to estimate eigenvalues of covariance operators associated with limiting Gaussian processes. In particular, we introduce the Rayleigh-Ritz method for approximating the largest eigenvalue and, for the first time, apply it in the context of statistical goodness-of-fit testing. This application focuses on limit distributions of statistics of the weighted L² type, particularly in distribution-free or classical testing scenarios, such as testing for exponentiality or normality in both univariate and multivariate data. Lastly, we demonstrate the application of these methods across various contexts, illustrating how accurate eigenvalue estimation can lead to an approximation of the limit distribution using the Pearson system, and how this influences efficiency assessments of the asymptotic Bahadur type.