On smoothness of Tukey depth contours

Abstract

Abstract. The smoothness of Tukey depth contours is a regularity condition often encountered in asymptotic theory, among others. This condition ensures that the Tukey depth fully characterizes the underlying multivariate probability distribution. In this paper we demonstrate that this regularity condition is rarely satisfied. It is shown that even well-behaved probability distributions with symmetrical, smooth and (strictly) quasiconcave densities may have non-smooth Tukey depth contours, and that the smoothness behaviour of depth contours is fairly unpredictable.

Halfspace depth and floating body

“…Despite being of critical importance, so far the only examples of distributions with smooth contours of hD are the (full-dimensional affine images of) α-symmetric distributions with α > 1, see Example 2. As discussed in Gijbels and Nagy [63], apart from those distributions, no other multivariate measure with smooth depth contours is known in statistics. In that paper, it is also shown that simple distributions such as mixtures of multivariate Gaußian distributions, and distributions with smooth centrally symmetric, or smooth strictly quasi-concave densities, may have points at which the boundary of P δ is not smooth.…”