  New inference methods for the multivariate coefficient of variation and its
reciprocal, the standardized mean, are presented. While there are various
testing procedures for both parameters in the univariate case, it is less known
how to do inference in the multivariate setting appropriately. There are some
existing procedures but they rely on restrictive assumptions on the underlying
distributions. We tackle this problem by applying Wald-type statistics in the
context of general, potentially heteroscedastic factorial designs. In addition
to the $k$-sample case, higher-way layouts can be incorporated into this
framework allowing the discussion of main and interaction effects. The
resulting procedures are shown to be asymptotically valid under the null
hypothesis and consistent under general alternatives. To improve the finite
sample performance, we suggest permutation versions of the tests and shown that
the tests' asymptotic properties can be transferred to them. An exhaustive
simulation study compares the new tests, their permutation counterparts and
existing methods. To further analyse the differences between the tests, we
conduct two illustrative real data examples.
