6 research outputs found
Joint Mixability of Elliptical Distributions and Related Families
In this paper, we further develop the theory of complete mixability and joint
mixability for some distribution families. We generalize a result of
R\"uschendorf and Uckelmann (2002) related to complete mixability of continuous
distribution function having a symmetric and unimodal density. Two different
proofs to a result of Wang and Wang (2016) which related to the joint
mixability of elliptical distributions with the same characteristic generator
are present. We solve the Open Problem 7 in Wang (2015) by constructing a
bimodal-symmetric distribution. The joint mixability of slash-elliptical
distributions and skew-elliptical distributions is studied and the extension to
multivariate distributions is also investigated.Comment: 15page
Current Open Questions in Complete Mixability
Complete and joint mixability has raised considerable interest in recent few
years, in both the theory of distributions with given margins, and applications
in discrete optimization and quantitative risk management. We list various open
questions in the theory of complete and joint mixability, which are
mathematically concrete, and yet accessible to a broad range of researchers
without specific background knowledge. In addition to the discussions on open
questions, some results contained in this paper are new
Extremal Dependence Concepts
The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent devel- opments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open ques- tions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative de- pendence given in literature and introduce a novel notion, which in a gen- eral sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. This is not only for historical reasons. Re- arrangement functions describe the relationship between random variables in a completely deterministic way, allow a deeper understanding of dependence itself, and have several advantages on the approximation of solutions in a broad class of optimization problems