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

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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.

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Acknowledgements

The authors thank Tobias Jahnke for providing the proof of Proposition 5 and the referees for helpful remarks.

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Authors and Affiliations

  1. Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, 76313, Karlsruhe, Germany

    Norbert Henze

  2. Department of Statistics, University of Pretoria (UP), Private Bag X20, Hatfield, 0028, South Africa

    Jaco Visagie

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  1. Norbert Henze
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  2. Jaco Visagie
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Correspondence to Norbert Henze.

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The second author’s work is based on research supported by the National Research Foundation, South Africa (Research chair: Non-parametric, Robust Statistical Inference and Statistical Process Control, Grant number 71199). Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the NRF.

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Henze, N., Visagie, J. Testing for normality in any dimension based on a partial differential equation involving the moment generating function. Ann Inst Stat Math 72, 1109–1136 (2020). https://doi.org/10.1007/s10463-019-00720-8

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  • DOI: https://doi.org/10.1007/s10463-019-00720-8

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