Studying mixability with supermodular aggregating functions

Bignozzi, Valeria; Puccetti, Giovanni

doi:10.1016/j.spl.2015.01.032

Cited by 6 publications

(7 citation statements)

References 14 publications

(24 reference statements)

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“…As special cases of (b), two extra conditions that can be easily checked, (c) length inequality and (d) variance inequality are also introduced. It is shown that any of (a)- Bignozzi and Puccetti (2015).…”

Section: Resultsmentioning

confidence: 99%

Joint Mixability

Wang

2016

Mathematics of OR

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Many optimization problems in probabilistic combinatorics and mass transportation impose fixed marginal constraints. A natural and open question in this field is to determine all possible distributions of the sum of random variables with given marginal distributions; the notion of joint mixability is introduced to address this question. A tuple of univariate distributions is said to be jointly mixable if there exist random variables, with respective distributions, such that their sum is a constant. We obtain necessary and sufficient conditions for the joint mixability of some classes of distributions, including uniform distributions, distributions with monotone densities, distributions with unimodal-symmetric densities, and elliptical distributions with the same characteristic generator. Joint mixability is directly connected to many open questions on the optimization of convex functions and probabilistic inequalities with marginal constraints. The results obtained in this paper can be applied to find extreme scenarios on risk aggregation under model uncertainty at the level of dependence.

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Section: Resultsmentioning

confidence: 99%

Joint Mixability

Wang

2016

Mathematics of OR

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“…As an illustration, in Figure 17 we show a set of three rearrangements under which the product of three Lognormal distributions is jointly mixable. A first step in this direction can be found in Bignozzi and Puccetti (2015).…”

Section: Conclusion and Future Research Directionsmentioning

confidence: 99%

Extremal Dependence Concepts

Puccetti¹,

Wang²

2015

Statist. Sci.

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106

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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 developments 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 questions 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 dependence given in literature and introduce a novel notion, which in a general 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. Rearrangement 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.Comment: Published at http://dx.doi.org/10.1214/15-STS525 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…For more details on the problems and a brief history of the concept of the mixability, we refer to the recent papers Puccetti, Wang and Wang (2012), Wang (2015) and Wang and Wang (2016). Bignozzi and Puccetti (2015) extend the concept of joint mixability and introduce the concept of φ-joint mixability.…”

Section: Introductionmentioning

confidence: 99%

“…where h, g are convex functions. In particular, if all α i are 1, then φ 1 becomes the sum operator and φ 2 becomes the product operator considered in Bignozzi and Puccetti (2015).…”

Section: Introductionmentioning

confidence: 99%

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The mixability of elliptical distributions with supermodular functions

Zhang¹,

Li²,

Yin³

2017

Preprint

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The concept of φ-complete mixability and φ-joint mixability was first introduced in Bignozzi and Puccetti (2015), which is a direct extension of complete and joint mixability. Following Bignozzi and Puccetti (2015), we consider two cases of φ and investigate the φ-joint mixability for elliptical distributions and logarithmic elliptical distributions. We obtain a necessary and sufficient condition for the φ-joint mixability of some distributions and a sufficient condition for uniqueness of the center of φ-joint mixability for some elliptical distributions.

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Studying mixability with supermodular aggregating functions

Cited by 6 publications

References 14 publications

Joint Mixability

Joint Mixability

Extremal Dependence Concepts

The mixability of elliptical distributions with supermodular functions

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