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). 1 Babić, 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.

Elliptical Symmetry Tests in \proglang{R}

“…Recently, Babić et al (2019) proposed a new test for elliptical symmetry both for specified and unspecified location. These tests are based on Le Cam's theory of asymptotic experiments and are optimal against generalized skew-elliptical alternatives, but they remain quite powerful under a much broader class of non-elliptical distributions.…”

“…Here, Σ Σ Σ is Tyler (1987)'s estimator of scatter and X X X is the sample mean. When the location is not specified, Babić et al (2019) propose tests that have a simple asymptotic chi-squared distribution under the null hypothesis of ellipticity, are affine-invariant, computationally fast, have a simple and intuitive form, only require finite moments of order 2, and offer much flexibility in the choice of the radial density f at which optimality (in the maximin sense) is achieved. Note that the Gaussian f is excluded, though, due to a singular information matrix; see Babić et al (2019).…”

“…When the location is not specified, Babić et al (2019) propose tests that have a simple asymptotic chi-squared distribution under the null hypothesis of ellipticity, are affine-invariant, computationally fast, have a simple and intuitive form, only require finite moments of order 2, and offer much flexibility in the choice of the radial density f at which optimality (in the maximin sense) is achieved. Note that the Gaussian f is excluded, though, due to a singular information matrix; see Babić et al (2019). We implemented in our package the test statistic based on the radial density f of the multivariate t distribution, multivariate power-exponential and multivariate logistic, though in principle any non-Gaussian choice for f is possible.…”

“…Given the omnipresence of the assumption of elliptical symmetry, it is essential to be able to test whether that assumption actually holds true or not for the data at hand. Numerous tests have been proposed in the literature, including Beran (1979), Baringhaus (1991), Koltchinskii and Sakhanenko (2000), Manzotti et al (2002), Schott (2002), Huffer and Park (2007), Cassart (2007) and Babić et al (2019). Tests for elliptical symmetry based on Monte Carlo simulations can be found in Diks and Tong (1999) and Zhu and Neuhaus (2000); Li et al (1997) recur to graphical methods and Zhu and Neuhaus (2004) build conditional tests.…”

“…This test assumes the location parameter to be known and its asymptotic distribution is not simple to use (plus no proven validity in dimensions higher than 2), hence we decided not to include it in the package. Thus, the tests suggested by Koltchinskii and Sakhanenko (2000), Manzotti et al (2002), Schott (2002), Huffer and Park (2007), Cassart (2007) and Babić et al (2019) are implemented in the package ellipticalsymmetry.…”