Choosing among notions of multivariate depth statistics

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.