A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657?658) of the d-dimensional data on the unit sphere of Rd. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.

[1]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[2]  R. C. Bose,et al.  Essays in probability and statistics , 1971 .

[3]  K C Rao,et al.  A chi-squabe statistic for goodies-of-fit tests within the exponential family , 1974 .

[4]  N. J. H. Small.  Plotting squared radii , 1978 .

[5]  Rudolf Beran,et al.  Testing for Ellipsoidal Symmetry of a Multivariate Density , 1979 .

[6]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[7]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[8]  T. W. Anderson,et al.  Statistical Inference in Elliptically Contoured and Related Distributions , 1990 .

[9]  Adolfo J. Quiroz,et al.  Some new tests for multivariate normality , 1991 .

[10]  Ludwig Baringhaus,et al.  Testing for Spherical Symmetry of a Multivariate Distribution , 1991 .

[11]  P. Bentler,et al.  A necessary test of goodness of fit for sphericity , 1993 .

[12]  Sigeo Aki,et al.  On nonparametric tests for symmetry inRm , 1993 .

[13]  S. Rachev,et al.  Testing Multivariate Symmetry , 1995 .

[14]  A. Quiroz,et al.  Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function , 1996 .

[15]  V. Koltchinskii,et al.  Testing for Spherical Symmetry of a Multivariate Distribution , 1998 .

[16]  R. Dudley,et al.  Uniform Central Limit Theorems: Notation Index , 2014 .

[17]  Spherical harmonics in quadratic forms for testing multivariate normality , 2001 .