  The M-estimators of multivariate scatter are known to have breakdown points
no greater than 1/(p+1), where p is the dimension of the data. In high
dimension, the breakdown points are usually considered to be disappointingly
low. This paper studies the breakdown problem in more detail. The exact
breakdown points for the M-estimators of scatter are obtained and it is shown
that their low values are primarily due to contamination restricted to some
plane. If such "coplanar" contamination is not present, then there exists
M-estimators which have breakdown points close to 1/2. The effect of coplanar
contamination is further examined and is shown to be related to the singularity
of the scatter matrix. Finally, the implications of the results of this paper
on whether the low breakdown point is necessarily a bad feature and on
multivariate outlier detection are briefly discussed.
  This paper is a reprint of an unpublished 1986 Rutgers Technical Report.
