# Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

@article{Drr2020TestingMN, title={Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces}, author={Philip D{\"o}rr and Bruno Ebner and Norbert Henze}, journal={Scandinavian Journal of Statistics}, year={2020}, volume={48}, pages={456 - 501} }

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…

## 8 Citations

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The 4th workshop on Goodness-of-fit, change-point, and related problems was held at the Department of Economics and Management of the University of Trento, September 6–8, 2019. The workshop followed…

## References

SHOWING 1-10 OF 83 REFERENCES

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

- MathematicsAnnals of the Institute of Statistical Mathematics
- 2019

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…

A class of invariant consistent tests for multivariate normality

- Mathematics
- 1990

Let be independent identically distributed random vectors in Rd d ≥ 1 , with sample mean [Xbar] n and sample covariance matrix S n . We present a class of practicable afflne-invariant tests for the…

A consistent test for multivariate normality based on the empirical characteristic function

- Mathematics
- 1988

AbstractLetX1,X2, …,Xn be independent identically distributed random vectors in IRd,d ⩾ 1, with sample mean
$$\bar X_n $$
and sample covariance matrixSn. We present a practicable and consistent test…

A New Approach to the BHEP Tests for Multivariate Normality

- Mathematics
- 1997

LetX1, ?, Xnbe i.i.d. randomd-vectors,d?1, with sample meanXand sample covariance matrixS. For testing the hypothesisHdthat the law ofX1is some nondegenerate normal distribution, there is a whole…

A new class of tests for multinormality with i.i.d. and garch data based on the empirical moment generating function

- Mathematics
- 2017

We generalize a recent class of tests for univariate normality that are based on the empirical moment generating function to the multivariate setting, thus obtaining a class of affine invariant,…

Testing normality via a distributional fixed point property in the Stein characterization

- MathematicsTEST
- 2019

We propose two families of tests for the classical goodness-of-fit problem to univariate normality. The new procedures are based on $$L^2$$ L 2 -distances of the empirical zero-bias transformation to…

Invariant tests for multivariate normality: a critical review

- Mathematics
- 2002

This paper gives a synopsis on affine invariant tests of the hypothesis that the unknown distribution of a d-dimensional random vector X is some nondegenerate d-variate normal distribution, on the…

Tests for multivariate normality based on canonical correlations

- MathematicsStat. Methods Appl.
- 2014

New affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution, are proposed, which are found to offer higher power against many of the alternatives.

CHARACTERIZATIONS OF MULTINORMALITY AND CORRESPONDING TESTS OF FIT, INCLUDING FOR GARCH MODELS

- MathematicsEconometric Theory
- 2018

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of…