The classical Lorenz curve visualizes and measures the disparity of items which are characterized by a single variable: The more the curve bends, the more scatter the data. Recently a general approach has been proposed and investigated that measures the disparity of multidimensioned items regardless of their dimension.

Multivariate Lorenz dominance based on zonoids

The paper introduces an approach to the ordering of dependence which is based on central regions. A d-variate probability distribution is described by a nested family of sets, called central regions. Those regions are affine equivariant, compact and starshaped and concentrate about a properly defined center. They can be seen as level sets of a depth function. Special cases are Mahalanobis, zonoid, and likelihood regions. A d-variate distribution is called more dependent than another one if the volume of each central region is smaller with the first distribution. This dependence order is characterized by an inequality between determinants of certain parameter matrices if either (i) F and G are arbitrary distributions and the central regions are Mahalanobis or (ii) F and G belong to an elliptical family of distributions and the central regions are arbitrary. If the regions are zonoid regions, the dependence order implies the ordering of lift zonoid volumes. Alternatively, the dependence order is applied to the copulae of the given distributions. Generalized correlation indices are proposed which are increasing with the dependence orders.

Central Regions and Dependency

A multivariate affine invariant family of depth-based tests is proposed for the one-sample location problem. Suitable outlyingness functions which are formulated using depth functions are used to construct the proposed tests. The asymptotic null distribution and the asymptotic relative efficiency of the tests are discussed under the class of centrally and elliptically symmetric distributions, respectively. Furthermore, a conditional distribution-free property of the tests is shown. The performance of the proposed tests is evaluated using a Monte Carlo study as well as asymptotic relative efficiencies and is compared to that of several competitors. It is observed that such tests yield a better performance as compared to their competitors for a wide spectrum of alternatives.

Affine invariant depth-based tests for the multivariate one-sample location problem

This thesis builds a framework for evaluating downside risk from multivariate data via a special class of risk measures (RM). The peculiarity of the analysis lies in getting rid of strong data distributional assumptions and in orientation towards the most critical data in risk management: those with asymmetries and heavy tails. At the same time, under typical assumptions, such as the ellipticity of the data probability distribution, the conformity with classical methods is shown. The constructed class of RM is a multivariate generalization of the coherent distortion RM, which possess valuable properties for a risk manager. The design of the framework is twofold. The first part contains new computational geometry methods for the high-dimensional data. The developed algorithms demonstrate computability of geometrical concepts used for constructing the RM. These concepts bring visuality and simplify interpretation of the RM. The second part develops models for applying the framework to actual problems. The spectrum of applications varies from robust portfolio selection up to broader spheres, such as stochastic conic optimization with risk constraints or supervised machine learning.

Geometrical Methods in Multivariate Risk Management: Algorithms and Applications

We define the β-skeleton depth based on the probability that a point is contained within the β-skeleton influence region of two i.i.d. random vectors. The proposed family of depth functions...

β-Skeleton depth functions and medians

Tests which are based on the rank statistic play an important role in univariate nonparametric statistics. The rank statistic of a sample assigns to each sample variable its position in the order statistic. If one tries to generalize rank tests to the multivariate case, one is faced with the problem of defining a multivariate order statistic. Because of the absence of a natural linear order on ℝ d it is not clear how to define a multivariate order or rank statistic in a meaningful way.

Inference based on data depth by Rainer Dyckerhoff

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Inference based on data depth by Rainer Dyckerhoff