  The multivariate contaminated normal (MCN) distribution represents a simple
heavy-tailed generalization of the multivariate normal (MN) distribution to
model elliptical contoured scatters in the presence of mild outliers, referred
to as "bad" points. The MCN can also automatically detect bad points. The price
of these advantages is two additional parameters, both with specific and useful
interpretations: proportion of good observations and degree of contamination.
However, points may be bad in some dimensions but good in others. The use of an
overall proportion of good observations and of an overall degree of
contamination is limiting. To overcome this limitation, we propose a multiple
scaled contaminated normal (MSCN) distribution with a proportion of good
observations and a degree of contamination for each dimension. Once the model
is fitted, each observation has a posterior probability of being good with
respect to each dimension. Thanks to this probability, we have a method for
simultaneous directional robust estimation of the parameters of the MN
distribution based on down-weighting and for the automatic directional
detection of bad points by means of maximum a posteriori probabilities. The
term "directional" is added to specify that the method works separately for
each dimension. Mixtures of MSCN distributions are also proposed as an
application of the proposed model for robust clustering. An extension of the EM
algorithm is used for parameter estimation based on the maximum likelihood
approach. Real and simulated data are used to show the usefulness of our
mixture with respect to well-established mixtures of symmetric distributions
with heavy tails.
