On some generalized medians

Date of Completion

January 1994






Two generalizations of the median in several dimensions are examined namely, Tukey's half space median and Liu's simplicial median. It is proved that both medians are asymptotically normal (for Tukey's median, this was proved by Nolan in the symmetric case on $R\sp2).$ Their robustness is also studied by means of breakdown points. It is shown that the breakdown point of the simplicial median is always positive (although it may not be too large), and the results of Donoho and Gasko on the breakdown point of Tukey's median are extended to the non-symmetric case. Other measures of robustness are also considered.^ The main conclusions of this work are: (1) the central limit theorem holds for Liu's sample simplicial median under symmetry and smoothness conditions, in particular, it is a $n\sp{1\over 2}$-consistent estimator of Liu's median of the population; (2) the central limit theorem holds on $R\sp2$ for Tukey's sample median under smoothness conditions, even if symmetry is not assumed; (3) Liu's simplicial median is robust but not as robust as Tukey's median. As a consequence, the consistency properties of Tukey's median are improved, and Liu's median added to the short list of multidimensional location parameters and estimators which are coordinate free, affine equivariant, robust and $n\sp{1\over 2}$ consistent. ^