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

Abstract

This paper deals with testing for nondegenerate normality of a d-variate random vector X based on a random sample X 1 ,. .. , X n of X. The rationale of the test is that the characteristic function ψ(t) = exp(− t 2 /2) of the standard normal distribution in R d is the only solution of the partial differential equation Δ f (t) = (t 2 −d) f (t), t ∈ R d , subject to the condition f (0) = 1, where Δ denotes the Laplace operator. In contrast to a recent approach that bases a test for multivariate normality on the difference Δψ n (t)− (t 2 − d)ψ(t), where ψ n (t) is the empirical characteristic function of suitably scaled residuals of X 1 ,. .. , X n , we consider a weighted L 2-statistic that employs Δψ n (t) − (t 2 − d)ψ 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.

Stein's Method Meets Statistics: A Review of Some Recent Developments

“…related to characteristic functions) are used. Dörr et al (2021) also introduce a test of multivariate normality, based on (T g)(x) = −∆g(x) + ( x 2 2 − d)g(x) (where ∆ denotes the Laplacian), and the class of test functions {g t (x) = exp(it x) : t ∈ R d }.…”

BHEP Class of Tests for Multivariate Normality: An Empirical Comparison

“…, subject to the condition (0) 1 f  . Based on this characterization, Dorr et al [29] obtained a statistic for assessing MVN of datasets. The statistic is given by:…”

“…The affine invariant and consistent test rejects the H 0 of MVN for large values of , Dorr et al[29] established as a theorem that the characteristic function of a d-variate normal distribution is the only solution of the partial differential equation…”

A New Goodness of Fit Test for Multivariate Normality and Comparative Simulation Study

“…Based on the outcome of normality verification, one can choose suitable analysis methods (parametric or non-parametric) for further investigation. From the end of the 20th century to the present day, multivariate tests for testing the goodness of fit hypothesis have been developed by a number of authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Some of the most popular and commonly used multivariate tests are Chi-Square [8], Cramer von Mises [2], Anderson-Darling [2], and Royston [3].…”

“…While for the univariate tests, the invariance property is always satisfied. The properties of invariance, contingency are presented in Section 2 and are discussed in more detail in [2,12,15].…”