Multidimensional medians and uniqueness

Multidimensional medians induced from depth functions as the generalizations of the univariate median have been proposed and studied. Like their univariate counterpart, they usually possess the desirable properties including affine equivariance, high breakdown point robustness, etc. Furthermore, they could serve as the deepest point (a location measure) of the underlying distribution. The most prominent and prevail depth median is Tukey’s halfspace median. However, like most of other depth medians, it is generally not unique. On the other hand, we show that the projection median distinguishes itself from its competitors and possesses the desirable uniqueness property.