Power-divergence copulas: A new class of Archimedean copulas, with an insurance application

TL;DR

This work reveals that the convex generators driving $\phi$-divergences can also generate Archimedean copulas, enabling the construction of power-divergence (PD) copulas tied to the well-known power divergences indexed by $\lambda$. The PD family exhibits rich dependence properties, including a continuous spectrum from Fréchet bounds to singular mass on a zero-curve, monotone negative ordering in $\lambda$, and distinctive tail-dependence behavior, with higher-dimensional validity established for select $\lambda$ values. The authors develop computational routines to evaluate and simulate from $C_{\lambda}$, and demonstrate practical utility by fitting a PD copula to Danish fire insurance data where competing copulas fail. They show that PD copulas can capture both upper-tail dependence and a nontrivial zero-curve structure, providing a flexible alternative for actuarial and multivariate dependence modelling. The work also points to broader opportunities to link other $\phi$-divergence generators to Archimedean copulas, advancing both theory and applications in risk and dependence modelling.

Abstract

This paper demonstrates that, under a particular convention, the convex functions that characterise the phi divergences also generate Archimedean copulas in at least two dimensions. As a special case, we develop the family of Archimedean copulas associated with the important family of power divergences, which we call the power-divergence copulas. The properties of the family are extensively studied, including the subfamilies that are absolutely continuous or have a singular component, the ordering of the family, limiting cases (i.e., the Frechet-Hoeffding lower bound and Frechet-Hoeffding upper bound), the Kendall's tau and tail-dependence coefficients, and cases that extend to three or more dimensions. In an illustrative application, the power-divergence copulas are used to model a Danish fire insurance dataset. It is shown that the power-divergence copulas provide an adequate fit to the bivariate distribution of two kinds of fire-related losses claimed by businesses, while several benchmarks (a suite of well known Archimedean, extreme-value, and elliptical copulas) do not.

Figures (8)

• Figure 1: Plots of $C_\lambda(u_1, u_2)$ for $\lambda \in \{-\sqrt{2},\sqrt{2}\}$ and $u_1, u_2 \in [0, 1]$ obtained numerically.
• Figure 2: For $\lambda = -\sqrt{2}$ (left panel) and $\lambda = \sqrt{2}$ (right panel), plots of the fourth root of $\partial^2C_\lambda(u_1, u_2)/\partial u_1\partial u_2$ for $u_1, u_2 \in [0, 1]$. The fourth root was used for display purposes only.
• Figure 3: Scatterplots of 1000 bivariate realisations from the power-divergence copulas, with $\lambda \in \{-10, -3, -2, -0.5, 1, 2, 3, 10\}$.
• Figure 4: Losses to profits versus total material losses (the sum of losses to building and contents) in millions of Danish krone (MDK) (left panel) and as copula data (right panel).
• Figure 5: (a) Contours of the empirical copula (jagged black lines) and fitted power-divergence (PD) copula, Galambos copula, and Tawn copula (see legend). The contours are displayed for the level curves of the copulas at $\{0.05, 0.15, ..., 0.85, 0.95\}$. (b) The copula data superimposed on (the fourth root of) the copula density of the fitted PD copula. (The fourth-root transformation of the density is purely intended to aid visualisation.)

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: e.g., Cressie2002
• Proposition 2.1
• Definition 2.5
• Proposition 3.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3