Abstract
This paper deals with testing for nondegenerate normality of a d-variate random vector X based on a random sample X_1,ldots ,X_n of X. The rationale of the test is that the characteristic function psi (t) = exp (-Vert tVert ^2/2) of the standard normal distribution in {mathbb {R}}^d is the only solution of the partial differential equation varDelta f(t) = (Vert tVert ^2-d)f(t), t in {mathbb {R}}^d, subject to the condition f(0) = 1, where varDelta denotes the Laplace operator. In contrast to a recent approach that bases a test for multivariate normality on the difference varDelta psi _n(t)-(Vert tVert ^2-d)psi (t), where psi _n(t) is the empirical characteristic function of suitably scaled residuals of X_1,ldots ,X_n, we consider a weighted L^2-statistic that employs varDelta psi _n(t) -(Vert tVert ^2-d)psi _n(t). We derive asymptotic properties of the test under the null hypothesis and alternatives. The test is affine invariant and consistent against general alternatives, and it exhibits high power when compared with prominent competitors. The main difference between the procedures are theoretically driven by different covariance kernels of the Gaussian limiting processes, which has considerable effect on robustness with respect to the choice of the tuning parameter in the weight function.
Highlights
A useful tool for assessing the fit of data to a family of distributions are empirical counterparts of distributional characterizations
Such differential equations have been used to test for multivariate normality, see Dörr et al (2020), Henze and Visagie (2020), exponentiality, see Baringhaus and Henze (1991a), the gamma distribution, see Henze et al (2012), the inverse Gaussian distribution, see Henze and Klar (2002), the beta distribution, see Riad and Mabood (2018), the univariate and multivariate skew-normal distribution, see Meintanis (2010) and Meintanis and Hlávka (2010), and the Rayleigh distribution, see Meintanis and Iliopoulos (2003)
The purpose of this paper is to investigate the effect on the power of a recent test for multivariate normality based on a characterization of the multivariate normal law in connection with the harmonic oscillator, see Dörr et al (2020)
Summary
Otto-von-Guericke-Universität Magdeburg, Institut für Mathematische Stochastik, Universitätsplatz 2, 39106 Magdeburg, Germany. The entries have been calculated by the method presented in Dörr et al (2020), Section 7, setting = 100, 000 and m = 2000 for d ∈ {2, 3, 5, 10} Note that this approach only relies on the structure of the covariance kernel given in Theorem 3(a), the multivariate normal distribution, and the weight function. The alternative distributions are selected to match those employed in the simulation studies in Dörr et al (2020), Henze and Visagie (2020), and are given as follows. We have only applied the methods as a proof of principle
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