  Classical multivariate statistics measures the outlyingness of a point by its
Mahalanobis distance from the mean, which is based on the mean and the
covariance matrix of the data. A multivariate depth function is a function
which, given a point and a distribution in d-space, measures centrality by a
number between 0 and 1, while satisfying certain postulates regarding
invariance, monotonicity, convexity and continuity. Accordingly, numerous
notions of multivariate depth have been proposed in the literature, some of
which are also robust against extremely outlying data. The departure from
classical Mahalanobis distance does not come without cost. There is a trade-off
between invariance, robustness and computational feasibility. In the last few
years, efficient exact algorithms as well as approximate ones have been
constructed and made available in R-packages. Consequently, in practical
applications the choice of a depth statistic is no more restricted to one or
two notions due to computational limits; rather often more notions are
feasible, among which the researcher has to decide. The article debates
theoretical and practical aspects of this choice, including invariance and
uniqueness, robustness and computational feasibility. Complexity and speed of
exact algorithms are compared. The accuracy of approximate approaches like the
random Tukey depth is discussed as well as the application to large and
high-dimensional data. Extensions to local and functional depths and
connections to regression depth are shortly addressed.
