  We introduce a class of regularized M-estimators of multivariate scatter and
show, analogous to the popular spatial sign covariance matrix (SSCM), that they
possess high breakdown points. We also show that the SSCM can be viewed as an
extreme member of this class. Unlike the SSCM, this class of estimators takes
into account the shape of the contours of the data cloud when down-weighing
observations. We also propose a median based cross validation criterion for
selecting the tuning parameter for this class of regularized M-estimators. This
cross validation criterion helps assure the resulting tuned scatter estimator
is a good fit to the data as well as having a high breakdown point. A
motivation for this new median based criterion is that when it is optimized
over all possible scatter parameters, rather than only over the tuned
candidates, it results in a new high breakdown point affine equivariant
multivariate scatter statistic.
