Invariant tests for the equality of k scale parameters under spherical symmetry

We consider tests for scale parameters when the underlying distribution belongs to the class of spherically symmetric laws. A (nx1) random vector x has a spherically symmetric distribution if the distribution of x is identical to the distribution of Px for all (n×n) orthogonal matrices P. Using the principle of invariance we show that the usual normal-theory tests are not only invariant tests but are also exactly robust with respect to this class of spherically symmetric laws.