  We propose generalizations of the Hotelling's $T^2$ statistic and the
Bhattacharayya distance for data taking values in Lie groups. A key feature of
the derived measures is that they are compatible with the group structure even
for manifolds that do not admit any bi-invariant metric. This property, e.g.,
assures analysis that does not depend on the reference shape, thus, preventing
bias due to arbitrary choices thereof. Furthermore, the generalizations agree
with the common definitions for the special case of flat vector spaces
guaranteeing consistency. Employing a permutation test setup, we further obtain
nonparametric, two-sample testing procedures that themselves are bi-invariant
and consistent. We validate our method in group tests revealing significant
differences in hippocampal shape between individuals with mild cognitive
impairment and normal controls.
