  Although the assumption of elliptical symmetry is quite common in
multivariate analysis and widespread in a number of applications, the problem
of testing the null hypothesis of ellipticity so far has not been addressed in
a fully satisfactory way. Most of the literature in the area indeed addresses
the null hypothesis of elliptical symmetry with specified location and actually
addresses location rather than non-elliptical alternatives. In this paper, we
are proposing new classes of testing procedures, both for specified and
unspecified location. The backbone of our construction is Le Cam's asymptotic
theory of statistical experiments, and optimality is to be understood locally
and asymptotically within the family of generalized skew-elliptical
distributions. The tests we are proposing are meeting all the desired
properties of a``good'' test of elliptical symmetry: they have a simple
asymptotic distribution under the entire null hypothesis of elliptical symmetry
with unspecified radial density and shape parameter; they are affine-invariant,
computationally fast, intuitively understandable, and not too demanding in
terms of moments. While achieving optimality against generalized
skew-elliptical alternatives, they remain quite powerful under a much broader
class of non-elliptical distributions and significantly outperform the
available competitors.
