On the planar spherical depth and lens depth

For a distribution function F on R^d and a point q ∈ R^d, the spherical depth SphD(q; F) is defined to be the probability that a point q is contained inside a random closed hyperball obtained from a pair of points from F. The lens depth LD(q; F) is defined analogously using hyperlens instead of hyperball in the definition of spherical depth. The spherical depth SphD(q; S) (lens depth LD(q; S)) is also defined for an arbitrary data set S ⊆ R^d and …