On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry

Abstract

Abstract Several robust nonparametric location estimators are examined with respect to several criteria, with emphasis on the criterion that they should agree with the point of symmetry in the case of a symmetric distribution. For this purpose, a broad version of multidimensional symmetry is introduced, namely “halfspace symmetry”, generalizing the well-known notions of “central” and “angular” symmetry. Characterizations of these symmetry notions are established, permitting their properties and interrelations to be illuminated. The particular location measures considered consist of several nonparametric notions of multidimensional median: The “L2” (or “spatial”), “Tukey/Donoho halfspace”, “projection”, and “Liu simplicial” medians, all of which are robust in the sense of nonzero breakdown point. It is established that the first three of these in general do identify the point of symmetry when it exists, whereas the latter, however, fails to do so in some circumstances. Combining this finding with consideration of other criteria such as affine equivariance, stochastic order preserving, and degree of robustness, we conclude that among these choices, the “halfspace” and “projection” medians, both of which are based on projection pursuit methodology, are the most attractive overall.