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The multivariate \(L_1\)-median and associated data depth. (English) Zbl 1054.62067

Summary: This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the \(L_1\)-median of a data cloud in \(\mathbb R^d\), a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; (ii) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different depth functions; and (iii) a simple closed-form formula of the \(L_1\)-depth function for a given data cloud in \(\mathbb R^d\).

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H99 Multivariate analysis
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