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

TL;DR

This paper introduces a Rayleigh-Ritz method to approximate eigenvalues of integral covariance operators for Hilbert-space valued Gaussian processes, aiding in statistical hypothesis testing and efficiency comparisons.

Contribution

It proposes a novel eigenvalue approximation technique using Rayleigh-Ritz, with applications to weighted L^2 statistics and goodness-of-fit tests.

Findings

Effective eigenvalue approximations for covariance operators

Application to critical value estimation in tests

Comparison of test efficiencies using approximations

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.