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

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 contrast 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 a Gaussian process. We solve the associated integral equation and present the corresponding eigenvalues.