A test for elliptical symmetry

• Published in: Journal of multivariate analysis Vol. 98; no. 2; pp. 256 - 281
• Main Authors: Huffer, Fred W., Park, Cheolyong
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.02.2007; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Bootstrap approximation; Chi-square test; Distribution theory; Elliptically contoured distribution; Elliptically contoured distribution Multivariate normality Chi-square test Bootstrap approximation; Exact sciences and technology; Hypotheses; Mathematics; Multivariate analysis; Multivariate normality; Probability and statistics; Sciences and techniques of general use; Simulation; Statistics; Studies; Symmetry; 62H15; Multivariate normality; Elliptically contoured distribution; Chi-square test; Bootstrap approximation; 62E20; Matrix analysis; Kurtosis measure; Chi square test; Multivariate analysis; Symmetric law; Statistical method; 62E20 Elliptically contoured distribution; Normality test; Skewness measure; Bootstrap
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2005.09.011

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].