On the eigenvalues associated with the limit null distribution of the Epps-Pulley test of normality

TL;DR

This paper analyzes the eigenvalues related to the limit null distribution of the Epps-Pulley test for normality, providing solutions to the integral equation involved.

Contribution

It solves the integral equation associated with the eigenvalues of the Epps-Pulley test's limit null distribution, advancing understanding of its theoretical properties.

Findings

Eigenvalues of the integral operator are explicitly computed.

The results enhance the theoretical foundation of the Epps-Pulley test.

The eigenvalues facilitate better understanding of the test's asymptotic behavior.

Abstract

The Shapiro--Wilk test (SW) and the Anderson--Darling test (AD) turned out to be strong procedures for testing for normality. They are joined by a class of tests for normality proposed by Epps and Pulley that, in contrary to SW and AD, have been extended by Baringhaus and Henze to yield easy-to-use affine invariant and universally consistent tests for normality in any dimension. The limit null distribution of the Epps--Pulley test involves a sequences of eigenvalues of a certain integral operator induced by the covariance kernel of the limiting Gaussian process. We solve the associated integral equation and present the corresponding eigenvalues.