We study a novel class of affine invariant and consistent tests for normality in any dimension in an i.i.d.-setting.The tests are based on a characterization of the standard d-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schrödinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure. K E Y W O R D Saffine invariance, consistency, empirical characteristic function, harmonic oscillator, neighborhood-of-model validation, test for multivariate normality INTRODUCTIONThe multivariate normal distribution plays a key role in classical and hence widely used procedures, such as multivariate linear regression models with fixed effects and multivari-This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G. This notion generalizes the classical notions of the arboricity and strong chromatic index.For a class F of graphs and a graph parameter p, let p(F) = sup{p(G) | G ∈ F}. We show that ia(F) is bounded from above by an absolute constant depending only on F, that is ia(F) = ∞ if and only if χ(F∇ 1 /2) = ∞, where F∇ 1 /2 is the class of 1 /2-shallow minors of graphs from F and χ is the chromatic number.Further, we give bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8 ≤ ia(F) ≤ 10.In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.
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.
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 ∆f (t) = ( t 2 − d)f (t), t ∈ R d , subject to the condition f (0) = 1. By contrast with a recent approach that bases a test for multivariate normality on the difference ∆ψ 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 ∆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.
We investigate extreme values of Mahonian and Eulerian distributions arising from counting inversions and descents of random elements in finite Coxeter groups. A triangular array of either of these distributions is constructed from a sequence of Coxeter groups with increasing ranks. We describe an appropriate scaling and variance constraints so that the row-wise maximum of this triangular array is attracted towards the Gumbel extreme value distribution.
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