Abstract.
The center of a univariate data set {x 1,…,x n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x 1−μ|,…,|x n−μ|). As the median and the mean are the minimizers of respectively the L 1- and the L 2-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x 1,…,x n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L 1-distances y′1=(∥x 1- μ∥1,…,∥x n- μ∥1) and the vector of L 2-distances y′2=(∥x 1- μ∥2,…,∥x n-μ∥2). We define the L 1-median and the L 1-mean as the minimizers of respectively the L 1- and the L 2-norm of y 1; and then the L 2-median and the L 2-mean as the minimizers of respectively the L 1- and the L 2-norm of y 2. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L 1-mean appears to be a new concept.
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Received January 1999
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Dodge, Y., Rousson, V. Multivariate L1 mean. Metrika 49, 127–134 (1999). https://doi.org/10.1007/s001840050029
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DOI: https://doi.org/10.1007/s001840050029