  We propose a class of locally and asymptotically optimal tests, based on
multivariate ranks and signs for the homogeneity of scatter matrices in $m$
elliptical populations. Contrary to the existing parametric procedures, these
tests remain valid without any moment assumptions, and thus are perfectly
robust against heavy-tailed distributions (validity robustness). Nevertheless,
they reach semiparametric efficiency bounds at correctly specified elliptical
densities and maintain high powers under all (efficiency robustness). In
particular, their normal-score version outperforms traditional Gaussian
likelihood ratio tests and their pseudo-Gaussian robustifications under a very
broad range of non-Gaussian densities including, for instance, all multivariate
Student and power-exponential distributions.
