Publication: Depth functions based on a number of observations of a random vector
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2007-04
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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 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.
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Convex hull, Depth function, Linear convex stochastic order, Multivariate Gini mean difference, Random set, Selection expectation, Simplicial depth, Sphere coverage