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Tails of multivariate Archimedean copulas
Arthur Charpentier, Johan Segers
TL;DR
The paper addresses the need for a complete account of upper and lower tails in multivariate Archimedean copulas, especially the varied structures associated with asymptotic independence. It builds a generator-based decision tree, evaluates 23 one-parameter models, and develops transformation methods for constructing tailored tails. The resulting directory supports classification and model construction while showing that asymptotic independence contains a rich range of tail dependence structures.
Problem
Existing work had not systematically characterized asymptotic independence in Archimedean copulas, although popular families exhibit it in at least one tail.
Method
The paper classifies both tails from computable generator characteristics, analyzes joint and survival asymptotics, and derives tail quantities for parametric and transformed generators.
Results
The decision tree assigns each upper and lower tail to one of three classes, and the taxonomy is applied to 23 one-parameter models.
Takeaways & Limitations
Asymptotic independence encompasses a surprisingly rich variety of tail dependence structures, while transformations enable models with tailor-made upper and lower tails.
Takeaways & Limitations
Even in simple cases, the authors could not identify the resulting limit distribution’s copula or survivor copula, or place either in a known copula family.
Abstract
from arXiv · showhide
A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.
1 Introduction
The paper studies the tail behaviour of Archimedean copulas, focusing on joint and survival probabilities and conditional distributions when components approach their extremes. It develops a taxonomy because asymptotic independence, common in applications, contains diverse tail structures that require more precise characterization.
- Motivation: Archimedean copulas model dependence separately from margins and are widely used in insurance, finance, hydrology, and survival analysis.They are defined through a decreasing convex generator and its generalized inverse; complete monotonicity also permits a frailty-model interpretation.
- Scope: The framework applies to multivariate random vectors whose dependence is represented by an Archimedean copula, while mixtures and hierarchical Archimedean models are outside scope.Tail knowledge is relevant to modeling joint occurrence of component extremes.
- Contribution: The paper classifies upper and lower tail dependence into asymptotic dependence and asymptotic independence categories.In asymptotic dependence, tail behaviour is governed by the generator’s regular-variation index near 0 or 1; asymptotic independence requires more detailed analysis.
- Contribution: The paper addresses an underexplored gap by giving a systematic account of the varied tail structures hidden within asymptotic independence.The authors note that popular parametric families exhibit asymptotic independence in at least one tail.
- Approach: The analysis examines joint and survival distributions and conditional distributions when selected components simultaneously approach 0 or 1.These results support derivation of domains of attraction, tail dependence copulas, and related tail quantities.
2 Overview and examples
The overview organizes both tails into three categories and provides practical routes for classifying models and constructing generators with selected tail behaviour. It applies the taxonomy to 23 one-parameter models and extends it through transformation families.
- Taxonomy: Each upper and lower tail is assigned to one of three categories using generator characteristics summarized by the decision tree.One category represents asymptotic dependence and two represent asymptotic independence.
- Examples: The taxonomy is applied to 23 one-parameter Archimedean generator models, with outcomes sometimes depending on the parameter value.A new model is added because a rare boundary case is absent from the first 22 models.
- Transformations: Five transformation families generate new Archimedean generators whose relevant tail quantities can, where possible, be expressed through those of a base generator.The transformations act either on generator values through convex increasing bijections or on arguments through concave increasing bijections.
- Transformations: Combining transformations yields multi-parameter families that include the Clayton, Gumbel, and Frank families.
3 Lower tail
The lower tail is classified by the generator’s behavior near zero: θ0 > 0 yields asymptotic dependence, whereas θ0 = 0 yields asymptotic independence with several distinct structures.
- θ0 > 0 gives lower-tail asymptotic dependence, while θ0 = 0 gives asymptotic independence for every variable pair.
- Asymptotic dependence: When θ0 > 0, simultaneous smallness of all variables has the same order as a single-variable event, and conditional limits can be computed.
- Asymptotic dependence: The resulting conditional limit distribution has the Clayton copula with parameter θ0.
- Asymptotic independence: non-strict generators: For non-strict generators, θ0 = 0 can imply zero probability of pairwise simultaneous smallness below a positive threshold.
- Asymptotic independence: strict generators: The precise-rate results require an additional regular-variation assumption, verified for all listed models with φ(0) = ∞ and θ0 = 0.
- Asymptotic independence: strict generators: For strict generators with θ0 = 0, the joint lower-tail probability is of precise order φ←(|I|φ(s)), under regular variation of ψ with finite index κ.
- Asymptotic independence: strict generators: The index κ determines tail heaviness relative to independence: κ = 0 gives ηL = 1/2, positive κ gives a heavier tail, and negative κ a lighter one.
- Asymptotic independence: strict generators: After normalization by χs, variables outside the conditioned set are of larger order and have independent limiting marginals, a strong form of asymptotic independence.
4 Upper tail
Upper-tail behavior is classified by the generator near 1: θ1 > 1 gives asymptotic dependence, while θ1 = 1 gives asymptotic independence with distinct subcases. The boundary case φ′(1) = 0 and θ1 = 1 yields especially rich conditional tail structures.
- 4 Upper tail: When θ1 = 1, every pair is asymptotically independent, and the joint upper-tail behavior depends on whether φ′(1) is negative or zero.The paper calls these near independence and near asymptotic dependence, respectively.
- 4.1 Asymptotic dependence: θ1 > 1 produces upper-tail asymptotic dependence, with pairwise coefficient λU = 2 − 2^(1/θ1), interpreted as 1 when θ1 = ∞.Conditioning on one component being close to 1 brings all other components close to 1 as well.
- 4.1 Asymptotic dependence: The resulting limiting distributions can be difficult to identify: even in d = 2, the paper cannot express their copula or survivor copula in a known family.This limitation is stated for the distribution arising in the asymptotic-dependence analysis and similarly for the near-asymptotic-dependence case.
- 4.2 Asymptotic independence: Near independence: If φ′(1) < 0, joint survivor probabilities can be proportional to those of the independence copula, giving a particularly strong form of asymptotic independence.This conclusion requires differentiability and a finite, positive derivative condition on φ← at zero.
- 4.3 Asymptotic independence: Near asymptotic dependence: If φ′(1) = 0 and θ1 = 1, conditioning on U1 > 1 − s makes the other variables converge to 1 more slowly than s, while becoming comonotone among themselves.The paper identifies this as a boundary between asymptotic independence and dependence.
5 Lower tail: Proofs
The lower-tail proofs reduce asymptotic behavior to regular variation of the generator and its inverse-related auxiliary functions. These arguments establish precise scaling and conditional-limit results for asymptotically independent lower tails.
- 5 Lower tail: Proofs: For φ(0) = ∞ and θ0 = 0, the auxiliary function ψ = −1/D(log φ←) is regularly varying with finite index κ satisfying κ ≤ 1.The proof derives this from ψ(φ(s)) = −sφ′(s) and the slowly varying behavior of φ.
- 5 Lower tail: Proofs: The proof establishes that the normalization χs gives the correct scale for variables outside the conditioned subset, which are of larger order than the conditioned variables.The resulting limiting conditional distribution has independent marginals.
6 Upper tail: Proofs
The upper-tail proofs derive limits by applying inclusion-exclusion, regular variation, and inverse-function arguments to φ near 1. The boundary case θ1 = 1 requires refined auxiliary functions to capture slower asymptotic rates.
- 6 Upper tail: Proofs: Inclusion-exclusion converts joint upper-tail events into expressions involving φ← evaluated at sums of φ(1 − sxi).Regular variation then yields the limiting dependence function for finite θ1.
- 6 Upper tail: Proofs: For φ′(1) = 0 and θ1 = 1, the proof introduces ℓ(s) = s^-1φ(1 − s) and ηs(x) = ℓ←(x^-1ℓ(s)) to control the conditional scale.Slow variation of ℓ implies ηs(x) tends to zero while s/ηs(x) tends to zero for 0 < x < 1.
- 6 Upper tail: Proofs: A second-order expansion uses auxiliary functions g and y(x,s) to obtain refined joint-tail asymptotics in the near-asymptotic-dependence case.The proof shows g is slowly varying and tends to zero under the stated assumptions.
A Regular variation of convex functions
Regular variation describes how a function scales near zero or infinity through a power-law index, with index zero corresponding to slow variation. For convex functions, this behavior can be characterized through derivative ratios.
- A Regular variation of convex functions: A function is regularly varying at zero with index τ when f(tx)/f(t) tends to x^τ as t decreases to zero; τ = 0 defines slow variation.The analogous definition at infinity uses t tending to infinity.
- A Regular variation of convex functions: For positive convex functions, regular variation is equivalent to a single derivative-ratio limit involving sf′(s)/f(s).This is the monotone density theorem used in the paper’s tail proofs.
- A Regular variation of convex functions: The proof obtains f(sx)/f(s) → x^τ by integrating the logarithmic derivative and taking the limit of its exponent.Convexity controls the derivative and enables the equivalence.
B Some useful formulas
This appendix develops integral identities for derivatives and uses them to derive a formula for multivariate Archimedean copulas through inclusion–exclusion and a change of variables.
- B.1: Lemma B.1 expresses finite differences of derivatives as iterated integrals involving (−D)^k f.The proof proceeds by induction, beginning with absolute continuity for k = 1 and applying the induction hypothesis to g(y) = f(y) − f(y + x_k).
- B.1: The induction step combines derivative differences and integrates over successive variables to obtain the stated formula.The argument uses the absolute continuity of D^{k−1}f and its Radon–Nikodym derivative D^k f.
- B.2: Lemma B.2 applies inclusion–exclusion to vectors u and v, restricting the summation to coordinates where u_j > 0.It defines Δ_j = φ(u_j) − φ(v_j), applies Lemma B.1, and changes variables y_j = φ(v_j) + t_j to derive equation (B.1).