Editor:
Universidad Carlos III de Madrid. Departamento de Estadística

Issued date:
2007-04

Serie/No.:
UC3M Working papers. Statistics and Econometrics 07-07

Keywords:
Convex hull
,
Depth function
,
Linear convex stochastic order
,
Multivariate Gini mean difference
,
Random set
,
Selection expectation
,
Simplicial depth
,
Sphere coverage

Rights:
Atribución-NoComercial-SinDerivadas 3.0 España

Abstract:

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 ofWe 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.[+][-]