Testing for normality in any dimension based on a partial differential equation involving the moment generating function

Abstract

We use a system of first-order partial differential equations that characterize the moment generating function of the d-variate standard normal distribution to construct a class of affine invariant tests for normality in any dimension. We derive the limit null distribution of the resulting test statistics, and we prove consistency of the tests against general alternatives. In the case d > 1, a certain limit of these tests is connected with two measures of multivariate skewness. The new tests show strong power performance when compared to well-known competitors, especially against heavy-tailed distributions, and they are illustrated by means of a real data set.

A new test of multivariate normality by a double estimation in a characterizing PDE

“…In the multivariate case, 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. Let NMix( p, μ, Σ) be the normal mixture distribution generated by where p ∈ (0, 1), μ ∈ R d , and Σ is a positive definite matrix.…”

“…These are denoted by S d (DIST), where DIST stands for the distribution of the radii, and was chosen to be the exponential, the beta and the χ 2 -distribution. Tables 3, 4, and 5 can be contrasted to Table 5-7 in Dörr et al (2020), and for n = 50, with Tables 3-5 in Henze and Visagie (2020). Again, we start with a comparison of T n,a and U n,a .…”

“…Again, U n,a seems to be much more sensitive to the choice of a proper tuning parameter than T n,a . Competing tests of multivariate normality are the Henze-Visagie test, see Henze and Visagie (2020), the Henze-Jiménez-Gamero test, see , the BHEP-test, the Henze-Jiménez-Gamero-Meintanis test, see , and the energy test, see Székely and Rizzo (2005). A description of the test statistics, as well as procedures for computing critical values is found in Henze and Visagie (2020).…”

“…Competing tests of multivariate normality are the Henze-Visagie test, see Henze and Visagie (2020), the Henze-Jiménez-Gamero test, see , the BHEP-test, the Henze-Jiménez-Gamero-Meintanis test, see , and the energy test, see Székely and Rizzo (2005). A description of the test statistics, as well as procedures for computing critical values is found in Henze and Visagie (2020). The BHEP-test performs best for the NMix (0.1, 3, I 2 )distribution (NMIX1 in Henze and Visagie 2020) but is outperformed by T n,a for NMix(0.5, 0, B 2 ), and by U n,a for the NMix(0.9, 3, B 2 ) (NMIX2 in Henze and Visagie 2020), where these procedures show the best performance of all tests considered.…”

“…Moreover, D denotes a differential operator, i.e., this operator can be regarded as ordinary differentiation if f is a function of only one variable or, for instance, the Laplace operator in the multivariate case. Such (partial) 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). In all these references, the authors propose a goodness-of-fit test by plugging in an empirical counterpart f n for f into ρ(D f , f ), and by measuring the deviation from the zero function in a suitable function space.…”