A note on simplicial depth function

We investigate the behaviour of simplicial depth under the perturbation (1−ε)F+ε δ z , where F is a p-dimensional probability distribution and δ z is the point-mass distribution concentrated at the point z. The influence function of simplicial depth at the point x, up to a scalar multiplier, turns out to be the difference between the conditional depth, given that one of the vertices of the random simplex is fixed at the position z, and the unconditional depth. The scalar multiplier is p+1, which suggests that simplicial depth can be more sensitive to perturbations as the dimensionality grows higher. The geometrical properties of the influence function give new insight into the observed behaviour of simplicial depth and its relation with halfspace depth. The behaviour of the perturbed simplicial median is also investigated.