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

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

A New Approach to the BHEP Tests for Multivariate Normality

AbstractLetX1, …, Xnbe i.i.d. randomd-vectors,d⩾1, with sample meanXand sample covariance matrixS. For testing the hypothesisHdthat the law ofX1is some nondegenerate normal distribution, there is a whole class of practicable affine invariant and universally consistent tests. These procedures are based on weighted integrals of the squared modulus of the difference between the empirical characteristic function of the scaled residualsYj=S−1/2(Xj−X) and its almost sure pointwise limit exp(−‖t‖2/2) underHd. The test statistics have an alternative interpretation in terms ofL2-distances between a nonparametric kernel density estimator and the parametric density estimator underHd, applied toY1, …, Yn. By working in the Fréchet space of continuous functions on Rd, we obtain a new representation of the limiting null distributions of the test statistics and show that the tests have asymptotic power against sequences of contiguous alternatives converging toHdat the raten−1/2, independent ofd

Tests for multivariate normality—a critical review with emphasis on weighted L2L^2-statistics

This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted L$^{2}$-statistics. Since weighted L$^{2}$-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt