Joint mixability and negative dependence

The series of working papers contains research results on complete mixability (introduced in WP01) and joint mixability (introduced in WP03). The notion of joint mixability, with complete mixability as a special case, leads to the most negative dependence structure among random variables concerning their aggregation, a well-known challenging topic (on the other hand, the most positive dependence, i.e., comonotonicity, is well understood). Joint mixability has wide applications in financial risk management, optimization, and operations research (see the review articles WP08 and WP09). Many results here are useful for the working paper series on Robust risk aggregation. Some basic terminology is listed below.

Some particular challenges and results are:

There are many unsolved mathematical questions (WP08). A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

Pairwise counter-monotonicity is studied systematically. This paper establishes a stochastic representation, an invariance property, an implication to negative association, an equivalence to joint mix, and a connection to quantile based risk sharing in WP02 of Risk sharing, allocation, and equilibria. The journal version is published in Insurance: Mathematics and Economics (2023).

This paper connects joint mix dependence (JM) and some classic negative dependence concepts (ND). In contrast to JM which solves a multi-marginal optimal transport problem, a combination of JM and ND solves a corresponding problem under uncertainty. This paper is cross-listed as WP03 of Optimal transport and its applications. The journal version is to appear in Mathematics of Operations Research (2024).

This paper contains results on the aggregation set for standard uniform marginals. The main results are a characterization of unimodal densities in this set (Theorem 1) and an identity of the aggregation set and the corresponding convex-order lower set for dimension more than 2 (Theorem 5), thus providing a small partial answer to Open Problem #12 of WP08. The journal version is published in Journal of Applied Probability (2019).

This paper applies results in this working paper series to multiple hypothesis testing. Main results in WP01, WP03 and WP07 as well as results from Robust risk aggregation are used to determine valid merging methods for p-values without assuming how these p-values are dependent. The journal version is published in Biometrika (2020).

The notion of conditional joint mixability is introduced to obtain analytical results on robust risk aggregation for dual utilities (i.e., distortion risk measures). This paper is cross-listed as WP13 of Robust risk aggregation. The journal version is published in Finance and Stochastics (2019).

The main result is that the center for n standard Cauchy distributions is not unique and the set of all centers is an interval [-nlog(n-1)/π, nlog(n-1)/π], thus providing an answer to Open Problem #1 of WP08. The journal version is published in Journal of Theoretical Probability (2019).

This paper offers a comprehensive review on extremal dependence concepts among random variables, including comonotonicity, countermonotonicity, mutual exclusivity, joint mixability, and a new notion of Σ-countermonotonicitiy. The journal version is published in Statistical Science (2015).

This survey paper on complete and joint mixability lists 12 open questions. Open Problem #1 is solved in WP10 and Open Problem #12 is partially solved in WP13. The journal version is published in Probability Surveys (2015).

This paper contains some of the most advanced results on joint mixability. In particular it offers a characterization for joint mixability of distributions with decreasing (or increasing) densities (Theorem 3.2), in addition to several results on other distributions and properties of joint mixability. The journal version is published in Mathematics of Operations Research (2016).

This paper contains several results on the asymptotic behaviour of the homogenous aggregation set. In particular, the normalized aggregation set converges to a convex-order lower set (Theorem 3.5). The journal version is published in Journal of Multivariate Analysis (2015).

This paper offers a numerical procedure to check joint mixability based on the Rearrangement Algorithm. The journal version is published in Journal of Computational and Applied Mathematics (2015).

The main results are complete mixability of distributions with a density bounded away from zero (Theorem 3.4) and an asymptotic equivalence formula for VaR/ES (Theorem 4.2, which will be superceded by results in WP06 of Robust risk aggregation). The journal version is published in Insurance: Mathematics and Economics (2013).

In this paper, the term joint mixability is coined. The main results are sharp bounds on the distribution function and VaR of aggregate risk with homogeneous marginals and monotone densities (Theorem 3.6 and Corollary 3.7). This paper is cross-listed as WP01 of Robust risk aggregation. The journal version is published in Finance and Stochastics (2013).

This paper contains several fundamental properties of complete mixability (Theorems 3.1 and 3.2) and a proof of the complete mixability of concave densities (Theorem 4.2). The journal version is published in Journal of Applied Probability (2012).

In this paper, the term complete mixability is coined. There are two main results. First, a distribution with monotone density is completely mixable if and only if the mean condition is satisfied (Theorem 2.4 and Corollary 2.9). Second, the smallest convex-order element of a homogeneous aggregation set with monotone densities is obtained (Theorem 3.4). The journal version is published in Journal of Multivariate Analysis (2011).