Operator Tail Densities for Copulas

• Operator tail densities for copulas are defined as density-level refinements that extend classical tail dependence to capture non-equivalent marginal tail decay.
• The methodology integrates operator regular variation through multivariate scaling and copula density decomposition to explicitly characterize limiting behavior, even for non-closed form copulas like Liouville distributions.
• This framework reveals hidden regular variation phenomena, unifies tail-dependence concepts, and provides actionable insights for modeling extremal dependence in multivariate risk.

Operator tail densities for copulas refine the classical theory of tail dependence in multivariate extremes by introducing operator-regular variation into the analysis of copula densities. This framework enables the decomposition of extremal dependence structures into components reflecting both the copula’s multivariate tail behavior and the heavy-tailed (or light-tailed) nature of the univariate margins. Operator tail densities unify and extend previous results on “tail density” for copulas, allowing for non-equivalent marginal tail rates and enabling explicit characterizations of limiting behavior—even for copulas that lack closed forms, such as those arising from Liouville distributions. The concept provides a rigorous density-level analogue of classical tail-dependence functions and exposes hidden regular variation phenomena.

1. Operator Regular Variation in Multivariate Densities

Operator regular variation generalizes scalar regular variation to multivariate settings by allowing for vector-valued scaling through a diagonal matrix of tail indices. For a non-negative random vector $X=(X_1,\dots,X_d)$ with density %%%%1%%%% on $\mathbb{R}_+^d$, fix a diagonal matrix $E = \mathrm{diag}(\lambda_1,\ldots,\lambda_d)$ with $\lambda_i>0$. The density $f$ is said to be multivariate regularly varying with operator index $E$, second index $\rho>0$, and limit density $\lambda(x)$, denoted $f \in \mathrm{MRV}(E,-\rho,\lambda(\cdot))$, if there exist scaling functions $g(t)=\mathrm{diag}(t^{\lambda_i}\ell_i(t))$, $\ell_i \in \mathrm{RV}_0$, and a scalar $V(t)\in\mathrm{RV}_{-\rho}$ such that

$$\frac{f(g(t)x)}{t^{-\mathrm{tr}(E)} V(t)} \longrightarrow \lambda(x), \quad t \to \infty, \quad x \in \mathbb{R}_+^d \setminus\{0\}$$

where $\mathrm{tr}(E) = \sum_i \lambda_i$. The limit $\lambda$ fulfills the quasi-homogeneity, $$\lambda(t^E x) = t^{-\rho-\mathrm{tr}(E)}\lambda(x)$$.

Each marginal $f_i$ is regularly varying: $f_i(t)\sim C_i t^{-\alpha_i-1}$ with $\alpha_i=\rho/\lambda_i$. The scaling breaks into "copula-part × marginal-RV-part," establishing a foundation for decomposing multivariate extreme value densities into marginal and dependence components (Li, 22 Dec 2025).

2. Operator Tail Density for Copulas

Consider a $d$-copula $C$ with joint density $c(u_1,\ldots,u_d)$ on $[0,1]^d$ and a vector of positive tail orders $\kappa=(\rho_1,\ldots,\rho_d)$. The copula is said to have an upper operator tail density of order $\kappa$ if there exist functions $r_i(u)\in\mathrm{RV}_{\rho_i}(0)$ and $\ell(u)\in\mathrm{RV}_0(0)$ such that, for each $w \in (0,\infty)^d$,

$$\lambda_C(w;\kappa) = \lim_{u\downarrow 0} \frac{c(1-r_1(u)w_1, \ldots, 1-r_d(u)w_d)}{u^{1-\sum_i\rho_i}\ell(u)}$$

This $\lambda_C$ satisfies the same quasi-homogeneity as the multivariate limit density:

$$\lambda_C(t^{\rho_1}w_1, \ldots, t^{\rho_d}w_d\,;\kappa) = t^{1-\sum_i\rho_i}\lambda_C(w;\kappa)$$

A lower tail density is defined analogously by applying the construction to the survival copula. This operator-level framework generalizes classical tail density (which often assumes the same marginal index) to cases of non-equivalent marginal tail decay, detecting hidden regular variation on sub-cones invisible to scalar indices (Li, 22 Dec 2025, Joe et al., 2019).

3. Decomposition Theorems: Copula and Margins to Operator Regular Variation

The main decomposition theorem states that the operator-regular variation of a full multivariate density $f$ can be characterized through the combination of the operator tail density of its copula and the regular variation of the marginals. Specifically:

• Suppose the copula $C$ has an upper operator tail density $\lambda_C(\cdot;\kappa)$, the marginals $f_i$ are regularly varying $f_i\in\mathrm{RV}_{-\alpha_i-1}$ ($\alpha_i>0$), and the scaling functions $r_i(u)$ are compatible with the marginal distributions via $r_i(u)\sim 1-F_i(u^{-1/\lambda_i})$ as $u\downarrow0$.
• With $g(t)=\mathrm{diag}(t^{\lambda_i})$ ($\lambda_i=\rho_i/\alpha_i$), the joint density $$f(x_1,\ldots,x_d) = c(F_1(x_1),\ldots,F_d(x_d))\prod_i f_i(x_i)$$ is operator-regularly varying with limit:

$$\lambda(x) = \lambda_C(x_1^{-\alpha_1},\ldots,x_d^{-\alpha_d};\kappa)\prod_{i=1}^d \alpha_i x_i^{-\alpha_i-1}$$

The limiting measure is thus a product of the copula’s operator tail density (evaluated on marginal power transforms) and the product of marginal power densities (Li, 22 Dec 2025).

4. Explicit Example: Liouville Copulas and Operator Tail Densities

Liouville copulas illustrate the utility of the operator tail density approach. A random vector $X\sim L_d[g; a_1, \ldots, a_d]$ with density proportional to $g(\sum_i x_i)\prod_i x_i^{a_i-1}$ and $g\in\mathrm{RV}_{-\beta}$, yields (for any choice of $\lambda_i>0$):

$$\frac{f(t^{\lambda_1}x_1,\ldots,t^{\lambda_d}x_d)}{t^{-\sum_i \lambda_i}V(t)} \longrightarrow \lambda(x) = \left(\sum_{i\in I_{\max}}x_i\right)^{-\beta} \prod_{i=1}^d x_i^{a_i-1}$$

with $I_{\max} = \{i: \lambda_i = \max_j \lambda_j\}$ and $$V(t) = g(t^{\max \lambda_j}) t^{\sum_i \lambda_i a_i}$$. The operator tail density of the Liouville copula $C$ (of order $\kappa=(1,\ldots,1)$) is:

$$\lambda_C(w_1, \ldots, w_d; (1,\ldots,1)) = \left(\sum_{i\in I_{\max}} w_i^{-1/\alpha_i}\right)^{-\beta} \prod_{i=1}^d \alpha_i^{-1} w_i^{-\big[(a_i-1)/\alpha_i + (\alpha_i+1)/\alpha_i\big]}$$

with $$\alpha_i=(\sum_j \lambda_j a_j - \beta \max \lambda_j)/\lambda_i$$. Notably, this explicit density is available even though the Liouville copula itself is not in closed form (Li, 22 Dec 2025).

5. Relation to Classical Tail-Dependence and Tail Measures

Operator tail densities provide a density-level refinement of the classical tail-dependence function. The (upper) tail-dependence function of order $\rho=(\rho_1, \ldots, \rho_d)$ for a copula $C$ is:

$$a_C(w; \rho) = \lim_{u\downarrow0} \frac{\mathbb{P}\{U_i > 1 - r_i(u)w_i \text{ for some } i\}}{u \ell(u)}$$

where $r_i, \ell$ match those in the definition of the operator tail density. Under regularity, there is the relation:

$$a_C(w; \rho) = \int_{\{x\ge0:\exists i: x_i > w_i\}} \lambda_C(x; \rho)\,dx$$

Hence, $\lambda_C$ directly yields the density of the limiting tail measure; its cumulative gives the classical tail-dependence function. Crucially, operator-indices $\rho_i$ permit variable marginal rates, capturing diverse joint tail risks and revealing hidden regular variation structures on lower-dimensional cones (Li, 22 Dec 2025).

6. Estimation and Practical Considerations

No specialized estimation method is proposed for operator tail densities. A pragmatic approach involves:

1. Transforming observations to the copula scale: $\widehat U_{ij} = 1 - \widehat F_i(X_{ij})$;
2. Selecting a small threshold $u>0$, collecting pseudo-observations with $\widehat U_{ij}<u$;
3. Performing density estimation (kernel or parametric) on $(w_{1j}, \ldots, w_{dj})$ with $w_{ij} = (1 - \widehat U_{ij})/r_i(u)$;
4. Regressing the empirical density against $u^{1-\sum \rho_i} \ell(u)$.

A rigorous mathematical treatment of estimation bias, threshold choice, and bandwidth selection for operator tail density estimation remains an open area (Li, 22 Dec 2025).

7. Connections to Skew-Elliptical Copula Tail Densities

Classical tail density analysis for copulas such as those arising from skew-elliptical distributions—skew-normal or skew-$t$—is subsumed as a special case. In these, the copula tail density typically takes the form:

$$\lambda_U(w; \kappa_U) = \lim_{u \to 0^+} \frac{c(1-u w)}{u^{\kappa_U - d} \ell_U(u)}$$

For heavy-tailed generator densities, the tail index and skewness strongly influence the form of $\lambda_U$, and all marginal indices are typically equal, so the operator tail density reduces to the scalar case (Joe et al., 2019). The operator tail density framework generalizes these results to allow for joint extremes with non-equivalent marginal behaviors.

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Summary Table: Key Elements in Operator Tail Density Theory

Concept	Definition/Formula	Reference
Operator regular variation	$$\displaystyle\frac{f(g(t)x)}{t^{-\operatorname{tr}(E)} V(t)} \to \lambda(x)$$	(Li, 22 Dec 2025)
Operator tail density of copula	$$\displaystyle\lambda_C(w;\kappa)=\lim_{u\downarrow0}\frac{c(1-r_1(u)w_1,\ldots)}{u^{1-\sum \rho_i}\ell(u)}$$	(Li, 22 Dec 2025)
Decomposition theorem (Theorem A)	$$\lambda(x) = \lambda_C(x_1^{-\alpha_1}, \ldots; \kappa)\,\prod_i \alpha_i x_i^{-\alpha_i-1}$$	(Li, 22 Dec 2025)
Liouville copula: operator tail dens.	Explicit formula for $\lambda_C(w_1,\ldots,w_d; (1,\ldots,1))$ as in Section 4	(Li, 22 Dec 2025)
Tail-dependence function and density	$$a_C(w; \rho) = \int_{\{x\ge0:\exists i: x_i > w_i\}} \lambda_C(x; \rho)\,dx$$	(Li, 22 Dec 2025)

The development and explicit characterization of operator tail densities for copulas provides a unified, density-level tool for analyzing extremal dependence in multivariate distributions—capable of accommodating non-equivalent marginal tail rates and directly subsuming classical tail-density and tail-dependence constructions.