A test for elliptical symmetry

Abstract

This paper presents a statistic for testing the hypothesis of elliptical symmetry. The statistic also provides a specialized test of multivariate normality. We obtain the asymptotic distribution of this statistic under the null hypothesis of multivariate normality, and give a bootstrapping procedure for approximating the null distribution of the statistic under an arbitrary elliptically symmetric distribution. We present simulation results to examine the accuracy of the asymptotic distribution and the performance of the bootstrapping procedure. Finally, for selected alternatives, we compare the power of our test statistic with that of recently proposed tests for elliptical symmetry given by Manzotti et al. [A statistic for testing the null hypothesis of elliptical symmetry, J. Multivariate Anal. 81 (2002) 274–285] and Schott [Testing for elliptical symmetry in covariance-matrix-based analyses, Statist. Probab. Lett. 60 (2002) 395–404], and with that of the well known tests for multivariate normality of Mardia [Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970) 519–530] and Baringhaus and Henze [A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1988) 339–348].