Paper: arXiv 2404.15023

Authors: Liyuan Lin, Ruodu Wang, Ruixun Zhang, Chaoyi Zhao

Abstract

We study the problem of choosing the copula when the marginal distributions of a random vector are not all continuous. Inspired by four motivating examples including simulation from copulas, stress scenarios, co-risk measures, and dependence measures, we propose to use the checkerboard copula, that is, intuitively, the unique copula with a distribution that is as uniform as possible within regions of flexibility. We show that the checkerboard copula has the largest Shannon entropy, which means that it carries the least information among all possible copulas for a given random vector. Furthermore, the checkerboard copula preserves the dependence information of the original random vector, leading to two applications in the context of diversification penalty and impact portfolios. The numerical and empirical results illustrate the benefits of using the checkerboard copula in the calculation of co-risk measures.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical analysis of checkerboard copulas for non-continuous marginals, proving maximum Shannon entropy properties. While predominantly theoretical, it includes empirical applications to co-risk measures and portfolio diversification that demonstrate practical utility.

Research Flowchart

  flowchart TD
    A[Research Goal: Choose optimal copula for non-continuous marginal distributions] --> B[Motivating Examples: Simulation, Stress Scenarios, Co-Risk & Dependence Measures]
    B --> C[Methodology: Construct Checkerboard Copula via uniformization]
    C --> D[Computational Process: Maximize Shannon Entropy across flexible regions]
    D --> E[Finding 1: Proves Checkerboard Copula carries minimal information while preserving original dependence]
    D --> F[Finding 2: Outperforms alternatives in diversification penalty, impact portfolios & co-risk calculations]

Cited References (51)

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  2. Risk exchange under infinite-mean Pareto models — → 2403.20171
  3. Optimal Impact Portfolios with General Dependence and Marginals (2024) — link
  4. Technical Note - An Unexpected Stochastic Dominance: Pareto Distributions, Dependence, and Diversification (2022) — link
  5. An Axiomatic Foundation for the Expected Shortfall (2020) — link
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  8. Testing for independence in arbitrary distributions (2019) — link
and 43 more
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  41. ii) Let V 2 ∼ U[0 , 1] be independent of ˆ X 2
  42. b) negatively regression dependent (NRD) if for every i ∈ [ d
  43. (i) Let ( V 1 , V 2 ) ∼ U (cid:0) [0 , 1] 2 (cid:1) be independent of ( ˆ X 1 , ˆ X 2 ). That is to use the checkerboard copula C ⊥

1 of 51 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.