The multivariate L₁-median and associated data depth

Abstract

This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the L₁-median of a data cloud in 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 dept functions; and (iii) a simple closed-form formula of the L₁-depth function for a given data cloud in ℝ^d.