On energy tests of normality

Abstract

The energy test of multivariate normality is an affine invariant test based on a characterization of equal distributions by energy distance. The test statistic is a degenerate kernel V-statistic, which asymptotically has a sampling distribution that is a Gaussian quadratic form under the null hypothesis of normality. The parameters of the limit distribution are the eigenvalues of the integral operator determined by the energy distance. Although a Monte Carlo approach provides excellent approximations to the sampling distribution of the test statistic for finite samples, in this work we develop two methods to obtain the eigenvalues and the asymptotic distribution of the energy test statistic. We derive the explicit integral equations for the eigenvalue problem for the simple and composite hypotheses of normality and solve them by a variation of Nyström’s method. For the simple hypothesis, we also obtain the eigenvalues by an empirical approach which we call the sample kernel method. Numerical results are summarized in tables of derived eigenvalues for several cases. The resulting probability distribution in each case is obtained by Imhof’s method. We also include simulation results that illustrate that for large samples the derived limit distribution is quite accurate in the upper tail. Software is available in the energy package for R to implement the tests by the original (Monte Carlo) approach and by applying the new methods using the derived asymptotic distribution.