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Title: Choosing among notions of statistical depth function Authors:  Karl Mosler - Universitaet zu Koeln (Germany)
Pavlo Mozharovskyi - Telecom Paris, Institut Polytechnique de Paris (France) [presenting]
Abstract: Classical statistics measures the outlyingness of a point by its Mahalanobis distance from the mean based on the covariance matrix of the data. Since the early 1990s, more general depth statistics have been developed for measuring centrality and outlyingness of multivariate data in a nonparametric way. A multivariate depth function is a function which, given a point and a distribution in d-dimensional space, yields a number between 0 and 1, while satisfying certain postulates regarding invariance, monotonicity, convexity and continuity. Accordingly, numerous notions of depth have been proposed in the literature, some of which are also robust against outlying data. The departure from classical Mahalanobis distance does not come without a cost. There is a trade-off between invariance, robustness and computational feasibility. Since recently, efficient exact and approximate algorithms for different depths and their applications have been made available in various software packages. In applications, there is a choice of a depth statistic: rather often various notions are feasible, among which we have to decide. Aspects and general principles of this choice are discussed. The speed of exact algorithms is compared. The limitations of popular approximate approaches are demonstrated, and guidelines are provided for the construction of depth-based statistical procedures, as well as for practical applications when several notions of depth appear to be computationally feasible.