Elliptical symmetry; Local asymptotic normality; Maximin tests; Multivariate skewness; Semiparametric inference; Skew-elliptical densities; Statistics and Probability
Abstract :
[en] 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.
Disciplines :
Mathematics
Author, co-author :
Babić, Slaðana; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium ; Vlerick Business School, Gent, Belgium
Gelbgras, Laetitia; Département de Mathématique, Université libre de Bruxelles, Bruxelles, Belgium
Hallin, Marc; ECARES, Département de Mathématique, Université libre de Bruxelles, Bruxelles, Belgium
LEY, Christophe ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Gent, Belgium
External co-authors :
yes
Language :
English
Title :
Optimal tests for elliptical symmetry: Specified and unspecified location
Publication date :
November 2021
Journal title :
Bernoulli
ISSN :
1350-7265
Publisher :
International Statistical Institute
Volume :
27
Issue :
4
Pages :
2189 - 2216
Peer reviewed :
Peer reviewed
Funding text :
Slad¯ana Babić was supported by the PhD Fellow grant 165880 of the Research Foundation-Flanders (FWO).
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