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

This paper deals with testing for nondegenerate normality of a d-variate random vector X based on a random sample $$X_1,\ldots ,X_n$$ X 1 , … , X n of X. The rationale of the test is that the characteristic function $$\psi (t) = \exp (-\Vert t\Vert ^2/2)$$ ψ ( t ) = exp ( - ‖ t ‖ 2 / 2 ) of the standard normal distribution in $${\mathbb {R}}^d$$ R d is the only solution of the partial differential equation $$\varDelta f(t) = (\Vert t\Vert ^2-d)f(t)$$ Δ f ( t ) = ( ‖ t ‖ 2 - d ) f ( t ) , $$t \in {\mathbb {R}}^d$$ t ∈ R d , subject to the condition $$f(0) = 1$$ f ( 0 ) = 1 , where $$\varDelta $$ Δ denotes the Laplace operator. In contrast to a recent approach that bases a test for multivariate normality on the difference $$\varDelta \psi _n(t)-(\Vert t\Vert ^2-d)\psi (t)$$ Δ ψ n ( t ) - ( ‖ t ‖ 2 - d ) ψ ( t ) , where $$\psi _n(t)$$ ψ n ( t ) is the empirical characteristic function of suitably scaled residuals of $$X_1,\ldots ,X_n$$ X 1 , … , X n , we consider a weighted $$L^2$$ L 2 -statistic that employs $$\varDelta \psi _n(t) -(\Vert t\Vert ^2-d)\psi _n(t)$$ Δ ψ 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.