A notion of multivariate median

JORGE G. ADROVER

Oslo, Norway

Conferencia; 25th European Meeting of Statistitians; 2005

European chapter of the Bernoulli Society for Mathematial Statistics and Probability (International Statistical Institute)

Multivariate medians have deserved lot of attention in the robustness literature since the seminal paper``Mathematics and the picturingof data´´ by J.W>Tukey, 1975. Since then, many notions of depth and multivariate medians have been introduced (see for instance, ``Multivariate analysis by data depth: Descriptive statistics, graphics and inference´´ by R. Liu, J. Parelius and K. Singh, Ann. Stat. 27, 1999, 783-858, or ``General Notions of Statistcal Depth´´ by Y. Zuo and R. Serfling, The Annals of Statistics, 28 (2), 461-482, 2000, for good accounts on the subject). We introduce a notion of multivariate median based on the fact that  a point deeply located in the data cloud should be close enough to convex sets containing one half of the data cloud. These ideas can be formalized to render a Fisher-consistent estimator in case of having a symmetric distribution. In the univariate case the estimator turns out to be the well known median. The proposal yields an estimator with attractive robustness properties whose asymptotic bias behavior is analyzed.