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Stochastic representation of Sarmanov copulas

Christopher Blier-Wong

TL;DR

This work transforms the admissibility problem for Sarmanov copulas into a constructive stochastic problem by representing bivariate Sarmanov copulas as Bernoulli-mixtures of independent margins, with the dependence captured by a latent pair $(I_1,I_2)$ and a simple relation $a=\Lambda_1\Lambda_2\theta$; this reduces the $d$-increasing checks to nonnegativity of a Bernoulli pmf. The method yields a unifying framework that recovers classical families such as the FGM and Huang–Kotz variants as special cases, and extends naturally to higher dimensions via a generalized FGM-type subset expansion with tractable parameter constraints and scalable simulation. Sharp global bounds for Spearman's rho and Kendall's tau are derived, and the same stochastic representation extends to powered and block-max transformed copulas, enabling easy construction of $C_{a,r}(u_1,u_2)=u_1u_2(1+ah_1(u_1)h_2(u_2))^r$. The approach separates marginal design from dependence design, supports exchangeable and non-exchangeable structures, and offers practical pathways for estimation and risk-modelling applications in high dimensions.

Abstract

Sarmanov copulas offer a simple and tractable way to build multivariate distributions by perturbing the independence copula. They admit closed-form expressions for densities and many functionals of interest, making them attractive for practical applications. However, the complex conditions on the dependence parameters to ensure that Sarmanov copulas are valid limit their application in high dimensions. Verifying the $d$-increasing property typically requires satisfying a combinatorial set of inequalities that makes direct construction difficult. To circumvent this issue, we develop a stochastic representation for bivariate Sarmanov copulas. We prove that every admissible Sarmanov can be realized as a mixture of independent univariate distributions indexed by a latent Bernoulli pair. The stochastic representation replaces the problem of verifying copula validity with the problem of ensuring nonnegativity of a Bernoulli probability mass function. The representation also recovers classical copula families, including Farlie--Gumbel--Morgenstern, Huang--Kotz, and Bairamov--Kotz--Bekçi as special cases. We further derive sharp global bounds for Spearman's rho and Kendall's tau. We then introduce a Bernoulli-mixing construction in higher dimensions, leading to a new class of multivariate Sarmanov copulas with easily verifiable parameter constraints and scalable simulation algorithms. Finally, we show that powered versions of bivariate Sarmanov copulas admit a similar stochastic representation through block-maximal order statistics.

Stochastic representation of Sarmanov copulas

TL;DR

This work transforms the admissibility problem for Sarmanov copulas into a constructive stochastic problem by representing bivariate Sarmanov copulas as Bernoulli-mixtures of independent margins, with the dependence captured by a latent pair and a simple relation ; this reduces the -increasing checks to nonnegativity of a Bernoulli pmf. The method yields a unifying framework that recovers classical families such as the FGM and Huang–Kotz variants as special cases, and extends naturally to higher dimensions via a generalized FGM-type subset expansion with tractable parameter constraints and scalable simulation. Sharp global bounds for Spearman's rho and Kendall's tau are derived, and the same stochastic representation extends to powered and block-max transformed copulas, enabling easy construction of . The approach separates marginal design from dependence design, supports exchangeable and non-exchangeable structures, and offers practical pathways for estimation and risk-modelling applications in high dimensions.

Abstract

Sarmanov copulas offer a simple and tractable way to build multivariate distributions by perturbing the independence copula. They admit closed-form expressions for densities and many functionals of interest, making them attractive for practical applications. However, the complex conditions on the dependence parameters to ensure that Sarmanov copulas are valid limit their application in high dimensions. Verifying the -increasing property typically requires satisfying a combinatorial set of inequalities that makes direct construction difficult. To circumvent this issue, we develop a stochastic representation for bivariate Sarmanov copulas. We prove that every admissible Sarmanov can be realized as a mixture of independent univariate distributions indexed by a latent Bernoulli pair. The stochastic representation replaces the problem of verifying copula validity with the problem of ensuring nonnegativity of a Bernoulli probability mass function. The representation also recovers classical copula families, including Farlie--Gumbel--Morgenstern, Huang--Kotz, and Bairamov--Kotz--Bekçi as special cases. We further derive sharp global bounds for Spearman's rho and Kendall's tau. We then introduce a Bernoulli-mixing construction in higher dimensions, leading to a new class of multivariate Sarmanov copulas with easily verifiable parameter constraints and scalable simulation algorithms. Finally, we show that powered versions of bivariate Sarmanov copulas admit a similar stochastic representation through block-maximal order statistics.
Paper Structure (18 sections, 11 theorems, 117 equations, 1 table)

This paper contains 18 sections, 11 theorems, 117 equations, 1 table.

Table of Contents

  1. Introduction
  2. Sarmanov copulas and a stochastic representation
  3. Sarmanov copulas
  4. Main result: stochastic representation for Sarmanov copulas
  5. Extremal Spearman's rho and Kendall's tau
  6. Tail dependence
  7. Multivariate extensions of Sarmanov copulas
  8. General multivariate construction
  9. Trivariate Sarmanov copulas
  10. Exchangeable Sarmanov copulas
  11. Multivariate orthant Spearman's rho and extremal dependence
  12. Power transformations of Farlie copulas
  13. Block maxima and outer-power transformations
  14. Bernoulli-mixture representation for integer powers
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 1

Assume $g_1$ and $g_2$ satisfy the absolute-continuity and bounded-derivative conditions stated in Section ss:farlie-def. Define scaling constants $\pi_1=\Lambda_1/(\Lambda_1 - \lambda_1)\in(0,1)$ and $\pi_2=\Lambda_2/(\Lambda_2-\lambda_2)\in(0,1).$ Define two auxiliary cdfs on $[0,1]$ for each marg Let $(U_{[0]},U_{[1]},V_{[0]},V_{[1]})$ be mutually independent, with $U_{[i]}\sim F_{U,[i]}$ and $

Theorems & Definitions (30)

  • Theorem 1
  • Example 1: FGM
  • Example 2: Huang--Kotz I
  • Example 3: Huang--Kotz II
  • Example 4: Bairamov--Kotz--Bekçi kernels
  • Remark 1
  • Proposition 1
  • proof
  • Theorem 2
  • Proposition 2
  • ...and 20 more