Depth functions based on a number of observations of a random vector

We present two statistical depth functions given in terms of the random variable defined as the minimum number of observations of a random vector that are needed to include a fixed given point in their convex hull. This random variable measures the degree of outlyingness of a point with respect to a probability distribution. We take advantage of this in order to define the new depth functions. Further, a technique to compute the probability that a point is included in the convex hull of a given number of i.i.d. random vectors is presented. Consider the sequence of random sets whose n-th element is the convex hull of $n$ independent copies of a random vector. Their sequence of selection expectations is nested and we derive a depth function from it. The relation of this depth function with the linear convex stochastic order is investigated and a multivariate extension of the Gini mean difference is defined in terms of the selection expectation of the convex hull of two independent copies of a random vector.