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November 2021 Optimal tests for elliptical symmetry: Specified and unspecified location
Slađana Babić, Laetitia Gelbgras, Marc Hallin, Christophe Ley
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Bernoulli 27(4): 2189-2216 (November 2021). DOI: 10.3150/20-BEJ1305


Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most papers in the literature indeed are dealing with the null hypothesis of elliptical symmetry with specified location and actually address location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam’s asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are enjoying all the desirable properties of a “good” test of elliptical symmetry: they have simple asymptotic distributions under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors.

Funding Statement

Slađana Babić was supported by the PhD Fellow grant 165880 of the Research Foundation-Flanders (FWO).


We thank the Editor and an anonymous referee for insightful comments which led to clarifying several points in the manuscript. We are grateful toYves Dominicy for sharing with us the daily returns dataset of Section 6.


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Slađana Babić. Laetitia Gelbgras. Marc Hallin. Christophe Ley. "Optimal tests for elliptical symmetry: Specified and unspecified location." Bernoulli 27 (4) 2189 - 2216, November 2021.


Received: 1 December 2019; Revised: 1 November 2020; Published: November 2021
First available in Project Euclid: 24 August 2021

Digital Object Identifier: 10.3150/20-BEJ1305

Keywords: Elliptical symmetry , local asymptotic normality , maximin tests , Multivariate skewness , semiparametric inference , skew-elliptical densities

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 4 • November 2021
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