  Data sets sampled in Lie groups are widespread, and as with multivariate
data, it is important for many applications to assess the differences between
the sets in terms of their distributions. Indices for this task are usually
derived by considering the Lie group as a Riemannian manifold. Then, however,
compatibility with the group operation is guaranteed only if a bi-invariant
metric exists, which is not the case for most non-compact and non-commutative
groups. We show here that if one considers an affine connection structure
instead, one obtains bi-invariant generalizations of well-known dissimilarity
measures: a Hotelling $T^2$ statistic, Bhattacharyya distance and Hellinger
distance. Each of the dissimilarity measures matches its multivariate
counterpart for Euclidean data and is translation-invariant, so that biases,
e.g., through an arbitrary choice of reference, are avoided. We further derive
non-parametric two-sample tests that are bi-invariant and consistent. We
demonstrate the potential of these dissimilarity measures by performing group
tests on data of knee configurations and epidemiological shape data.
Significant differences are revealed in both cases.
