On the use of robust estimators of multivariate location for heterogeneous data

The properties of orthogonal equivariance and componentwise increasingness are often discussed for estimators of multivariate location. The former property is linked to a coordinate-free nature of the data (as in spatial data), whereas the latter property is linked to an ordered nature of the data (typically formed by several univariate variables). Since both properties do not get along well with each other, if robustness in the presence of outliers is to be pursued, one must choose between them. Admittedly, most of the literature on multivariate estimation of location, in which affine equivariance is pursued, typically abandons the latter property. Unfortunately, in the more and more common presence of heterogeneous data (as in spatio-temporal data), both orthogonal equivariance (therefore, affine equivariance) and componentwise increasingness might bring very disappointing results. For this very reason, both properties should be forfeited and only required for some of the components.