Optimal tests for elliptical symmetry: Specified and unspecified location

Abstract

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 of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses 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 meeting all the desired properties of a"good" test of elliptical symmetry: they have a * Slađana Babić was supported by a grant (165880) as a PhD Fellow of the Research Foundation-Flanders (FWO). 1Babić, Gelbgras, Hallin, and Ley/Optimal tests for elliptical symmetry 2 simple asymptotic distribution 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.

The generalized hyperbolic family and automatic model selection through the multiple‐choice LASSO

“…To corroborate such a visual insight, we performed a test of elliptical symmetry. Among the various tests of elliptical symmetry available in the literature, we considered the MPQ test of Manzotti, Pérez, and Quiroz [27], which is implemented by the MPQ() function of the ellipticalsymmetry package [3]. We use this test because it preserves the claimed nominal significance level [2].…”

“…Among the various tests of elliptical symmetry available in the literature, we considered the MPQ test of Manzotti, Pérez, and Quiroz [27], which is implemented by the MPQ() function of the ellipticalsymmetry package [3]. We use this test because it preserves the claimed nominal significance level [2]. The resulting $$$ p $$$‐value is $$$ 0.032 $$$, suggesting the rejection of the null hypothesis of elliptical symmetry, the type of symmetry underlying the symmetric GH and NIG models, at the common $$$ 5\% $$$ significance level.…”

Visual tests for elliptically symmetric distributions

“…Batsidis et al (2014) proposed a family of power divergence‐type test statistics to test for elliptical symmetry. More recently, Bianco et al (2017) have proposed a testing technique for elliptical symmetry based on a robust location and scatter estimator, while Babic et al (2019) have proposed a testing procedure, based on Le Cam's asymptotic theory, for a family of generalized elliptically symmetric distributions. Albisetti et al (2020) proposed a testing procedure for spherical and elliptical symmetry based on the Kolmogorov‐Smirnov (KS) statistic.…”