Testing for Ellipsoidal Symmetry of a Multivariate Density

Let Z be a random vector whose distribution is spherically symmetric about the origin. A random vector X which is representable as the image of Z under affine transformation is said to have an ellipsoidally symmetric distribution. The model of ellipsoidal symmetry is a useful generalization of multivariate normality. This paper proposes and studies some goodness-of-fit tests which have good asymptotic power over a broad spectrum of alternatives to ellipsoidal symmetry.