586 research outputs found
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
Recommended from our members
On the worst and least possible asymptotic dependence
Multivariate extremes behave very differently under asymptotic dependence as compared to asymptotic independence. In the bivariate setting, we are able to characterise the extreme behaviour of the asymptotic dependent case by using the concept of the copula. As a result, we are able to identify the properties of the boundary cases, that are asymptotic independent but still have some asymptotic dependent features. These situations are the most problematic in statistical extreme, and, for this reason, distinguishing between asymptotic dependence and asymptotic independence represents a difficult problem. We propose a simple test to resolve this issue which is an alternative to the procedure based on the classical coefficient of tail dependence. In addition, we are able to identify the worst/least asymptotic dependence (in the presence of asymptotic dependence) that maximises/minimises the probability of a given extreme region if tail dependence parameter is fixed. It is found that the perfect extreme association is not the worst asymptotic dependence, which is consistent with the existing literature. We are able to find lower and upper bounds for some risk measures of functions of random variables. A particular example is the sum of random variables, for which a vivid academic effort has been noticed in the last decade, where bounds for a sum of random variables are sought. It is numerically shown that our approach provides a great improvement of the existing methods, which reiterates the sensible conclusion that any additional piece of information on dependence would help to reduce the spread of these bounds
Convolution Bounds on Quantile Aggregation
Quantile aggregation with dependence uncertainty has a long history in
probability theory with wide applications in finance, risk management,
statistics, and operations research. Using a recent result on inf-convolution
of quantile-based risk measures, we establish new analytical bounds for
quantile aggregation which we call convolution bounds. In fact, convolution
bounds unify every analytical result and contribute more to the theory of
quantile aggregation, and thus these bounds are genuinely the best one
available. Moreover, convolution bounds are easy to compute, and we show that
they are sharp in many relevant cases. Convolution bounds enjoy several other
advantages, including interpretability on the extremal dependence structure,
tractability, and theoretical properties. The results directly lead to bounds
on the distribution of the sum of random variables with arbitrary dependence,
and we illustrate a few applications in operations research
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
- …