A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

Abstract

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, Biometrika 65, 657-658) of the d-dimensional data on the unit sphere of R d . 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.

Estimating the tail-dependence coefficient: Properties and pitfalls

“…For instance, in the case of empirical marginal distributions and specific copula, we suggest to test the goodness-of-fit of the copula via (non-)parametric procedures such as those developed in Fermanian (2005), Chen et al (2003), Dobrić and Schmid (2004) or Genest et al (2005). Further, if one makes use of an elliptically contoured distribution, then we suggest to test for ellipticity; see, e.g., Manzotti et al (2002). However, we do not recommend a TDC estimation as presented in Section 3.4, since we do not know a suitable test for Grades rank from 1 to 6 with 1 excellent and 6 poor.…”

Extreme behavior of bivariate elliptical distributions

“…Primarily, these distributions allow an alternative to and extension of the normal law. Elliptical distributions are easily implemented and simulated (see, for example, Breymann et al (2003), Hodgson et al (2002), Johnson (1987), Li et al (1997), Manzotti et al (2002)), and they are useful for actuarial and financial applications.…”

Testing Group Symmetry of a Multivariate Distribution

“…A similar approach was used to test for multivariate normality in [16]. The authors of [6] developed a different kind of tests for ellipsoidal symmetry based on spherical harmonics. ), where γ(c) := m{v : v, u ≤ c}.…”