114 research outputs found
On the expected number of facets for the convex hull of samples
This paper studies the convex hull of -dimensional samples i.i.d.
generated from spherically symmetric distributions. Specifically, we derive a
complete integration formula for the expected facet number of the convex hull.
This formula is with respect to the CDF of the radial distribution. As the
number of samples approaches infinity, the integration formula enables us to
obtain the asymptotic value of the expected facet number for three categories
of spherically symmetric distributions. Additionally, the asymptotic result can
be applied to estimating the sample complexity in order that the probability
measure of the convex hull tends to one
Limit theorems for the diameter of a random sample in the unit ball
We prove a limit theorem for the maximum interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit d-dimensional ball for d≥2. The results are specialised for the cases when the points have spherical symmetric distributions, in particular, are uniformly distributed in the whole ball and on its boundary. Among other examples, we also give results for distributions supported by pointed sets, such as a rhombus or a family ofsegment
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 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.
Data depth and floating body
Little known relations of the renown concept of the halfspace depth for
multivariate data with notions from convex and affine geometry are discussed.
Halfspace depth may be regarded as a measure of symmetry for random vectors. As
such, the depth stands as a generalization of a measure of symmetry for convex
sets, well studied in geometry. Under a mild assumption, the upper level sets
of the halfspace depth coincide with the convex floating bodies used in the
definition of the affine surface area for convex bodies in Euclidean spaces.
These connections enable us to partially resolve some persistent open problems
regarding theoretical properties of the depth
Convex hulls of several multidimensional Gaussian random walks
We derive explicit formulae for the expected volume and the expected number
of facets of the convex hull of several multidimensional Gaussian random walks
in terms of the Gaussian persistence probabilities. Special cases include the
already known results about the convex hull of a single Gaussian random walk
and the -dimensional Gaussian polytope with or without the origin
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