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Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics

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  • Bruno Ebner

    (Karlsruhe Institute of Technology (KIT))

  • Norbert Henze

    (Karlsruhe Institute of Technology (KIT))

Abstract

This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted $$L^2$$ L 2 -statistics. Since weighted $$L^2$$ L 2 -statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt.

Suggested Citation

  • Bruno Ebner & Norbert Henze, 2020. "Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$ L 2 -statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 845-892, December.
    • Handle: RePEc:spr:testjl:v:29:y:2020:i:4:d:10.1007_s11749-020-00740-0
    DOI: 10.1007/s11749-020-00740-0
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