The expected convex hull trimmed regions of a sample

Given a data set in the multivariate Euclidean space, we study regions of central points built by averaging all their subsets with a fixed number of elements. The averaging of these sets is performed by appropriately scaling the Minkowski or elementwise summation of their convex hulls. The volume of such central regions is proposed as a multivariate scatter estimate and a circular sequence algorithm to compute the central regions of a bivariate data set is described

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.

THE EXPECTED CONVEX HULL TRIMMED REGIONS OF A SAMPLE

Given a data set in the multivariate Euclidean space, we study regions of central points built by averaging all their subsets with a fixed number of elements. The averaging of these sets is performed by appropriately scaling the Minkowski or elementwise summation of their convex hulls. The volume of such central regions is proposed as a multivariate scatter estimate and a circular sequence algorithm to compute the central regions of a bivariate data set is described.

Band depths based on multiple time instances

Bands of vector-valued functions $f:T\mapsto\mathbb{R}^d$ are defined by considering convex hulls generated by their values concatenated at $m$ different values of the argument. The obtained $m$-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for $m=1$) to the convex hull in the functional space if the number $m$ of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions.Comment: 12 page

MULTIVARIATE RISKS AND DEPTH-TRIMMED REGIONS

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.

Multivariate risk measures : a constructive approach based on selections

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. The definition of the selection risk measure is based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained risk measure is coherent (or convex), law invariant and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010), and Hamel et al. (2013)Supported by the Spanish Ministry of Science and Innovation Grants No. MTM20II—22993 and ECO20ll-25706. Supported by the Chair of Excellence Programme of the Universidad Carlos III de Madrid and Banco Santander and the Swiss National Foundation Grant No. 200021-13752

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationshi

Consistency of the α\alpha-trimming of a probability. Applications to central regions

The sequence of $\alpha$-trimmings of empirical probabilities is shown to converge, in the Painlev\'{e}--Kuratowski sense, on the class of probability measures endowed with the weak topology, to the $\alpha$-trimming of the population probability. Such a result is applied to the study of the asymptotic behaviour of central regions based on the trimming of a probability.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ109 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Multivariate expectile trimming and the BExPlot

Expectiles are the solution to an asymmetric least squares minimization problem for univariate data. They resemble some similarities with the quantiles, and just like them, expectiles are indexed by a level α. In the present paper, we introduce and discuss the main properties of the expectile multivariate trimmed regions, a nested family of sets, whose instance with trimming level α is built up by all points whose univariate projections lie between the expectiles of levels α and 1 − α of the projected dataset. Such trimming level is interpreted as the degree of centrality of a point with respect to a multivariate distribution and therefore serves as a depth function. We study here the convergence of the sample expectile trimmed regions to the population ones and the uniform consistency of the sample expectile depth. We also provide efficient algorithms for determining the extreme points of the expectile regions as well as for computing the depth of a point in R2. These routines are based on circular sequence constructions. Finally, we present some real data examples for which the Bivariate Expectile Plot (BExPlot) is introduced.This research was partially supported by the Spanish Ministry of Science and Innovation under grant ECO2015-66593-P

Data depth and multiple output regression, the distorted M-quantiles approach

For a univariate distribution, its M-quantiles are obtained as solutions to asymmetric minimization problems dealing with the distance of a random variable to a fixed point. The asymmetry refers to the different weights for the values of the random variable at either side of the fixed point. We focus on M-quantiles whose associated losses are given in terms of a power. In this setting, the classical quantiles are obtained for the first power, while the expectiles correspond to quadratic losses. The M-quantiles considered here are computed over distorted distributions, which allows to tune the weight awarded to the more central or peripheral parts of the distribution. These distorted M-quantiles are used in the multivariate setting to introduce novel families of central regions and their associated depth functions, which are further extended to the multiple output regression setting in the form of conditional regression regions and conditional depths