  We propose halfspace depth concepts for scatter, concentration and shape
matrices. For scatter matrices, our concept is similar to those from Chen, Gao
and Ren (2017) and Zhang (2002). Rather than focusing, as in these earlier
works, on deepest scatter matrices, we thoroughly investigate the properties of
the proposed depth and of the corresponding depth regions. We do so under
minimal assumptions and, in particular, we do not restrict to elliptical
distributions nor to absolutely continuous distributions. Interestingly, fully
understanding scatter halfspace depth requires considering different
geometries/topologies on the space of scatter matrices. We also discuss, in the
spirit of Zuo and Serfling (2000), the structural properties a scatter depth
should satisfy, and investigate whether or not these are met by scatter
halfspace depth. Companion concepts of depth for concentration matrices and
shape matrices are also proposed and studied. We show the practical relevance
of the depth concepts considered in a real-data example from finance.
