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  • Arthur CHARPENTIER - Archimedean copulas.Les copules Archimédiennes,quelques motivations et applicationsArthur CharpentierKatholieke Universiteit Leuven, ENSAE/CRESTInstitut de Mathématiques Appliquées, Angers, Novembre 20061
  • Arthur CHARPENTIER - Archimedean copulas.“Everybody who opens any journal on stochastic processes, probability theory,statistics, econometrics, risk management, finance, insurance, etc., observesthat there is a fast growing industry on copulas [...] The InternationalActuarial Association in its hefty paper on Solvency II recommends usingcopulas for modeling dependence in insurance portfolios [...] Since Basle IIcopulas are now standard tools in credit risk management”.“Are copulas suitable for modeling multivariate extremes? Copulas generateany multivariate distribution. If one wants to make an honest analysis ofmultivariate extremes the distributions used should be related to extremevalue theory in some way.” Mikosh (2005).2
  • Arthur CHARPENTIER - Archimedean copulas.“We are thus generally sympathetic to the primary objective pursued by Dr.Mikosch, which is to caution optimism about what copulas can and cannotachieve as a dependence modeling tool”.“Although copula theory has only recently emerged as a distinct field ofinvestigation, its roots go back at least to the 1940s, with the seminal work ofHoeőding on margin-free measures of association [...] “It was possiblyDeheuvels who, in a series of papers published around 1980, revealed the fullpotential of the fecund link between multivariate analysis and rank-basedstatistical techniques”.“However, the generalized use of copulas for model building (andArchimedean copulas in particular) seems to have been largely fuelled at theend of the 1980s by the publication of significant papers by Marshall andOlkin (1988) and by Oakes (1989) in the influential Journal of the AmericanStatistical Association”. Genest & Rémillard (2006).3 View slide
  • Arthur CHARPENTIER - Archimedean copulas.Definition 1. A 2-dimensional copula is a 2-dimensional cumulativedistribution function restricted to [0, 1]2with standard uniform margins.Copula (cumulative distribution function) Level curves of the copulaCopula density Level curves of the copula4 View slide
  • Arthur CHARPENTIER - Archimedean copulas.Why using copulas ?Theorem 2. (Sklar) Let C be a copula, and FX and FY two marginaldistributions, then F(x, y) = C(FX (x), FY (y)) is a bivariate distributionfunction, with F ∈ F(FX, FY ).Conversely, if F ∈ F(FX , FY ), there exists C such thatF(x, y) = C(FX(x), FY (y). Further, if FX and FY are continuous, then C isunique, and given byC(u, v) = F(F−1X (u), F−1Y (v)) for all (u, v) ∈ [0, 1] × [0, 1]We will then define the copula of F, or the copula of (X, Y ).5
  • Arthur CHARPENTIER - Archimedean copulas.In dimension 2, consider the following family of copulaeDefinition 3. Let ψ denote a convex decreasing function [0, 1] → [0, ∞] suchthat ψ(1) = 0. Define the inverse (or quasi-inverse if ψ(0) < ∞) asψ←(t) =ψ−1(t) for 0 ≤ t ≤ ψ(0)0 for ψ(0) < t < ∞.ThenC(u1, u2) = ψ←(ψ(u1) + ψ(u2)), u1, u2 ∈ [0, 1],is a copula, called an Archimedean copula, with generator ψ.Note that ψ←◦ ψ(t) = t on [0, 1]. ψ is said to be strict if ψ(0) = ∞.The generator is unique up to a multiplicative positive constant.6
  • Arthur CHARPENTIER - Archimedean copulas.In higher dimension, most of the notions and results can be extended.Definition 4. A d-dimensional copula is a d-dimensional cumulativedistribution function restricted to [0, 1]dwith standard uniform margins.Sklar’s theorem can be extended in dimension d as followsTheorem 5. (Sklar) Let C be a d-copula, and F1, ..., Fd be marginaldistributions, then F(x1, ..., , xd) = C(F1(x1), ..., Fd(xd)) is a d-dimensionaldistribution function, with F ∈ F(F1, ..., Fd).Conversely, if F ∈ F(F1, ..., Fd), there exists C such thatF(x1, ..., , xd) = C(F1(x1), ..., Fd(xd)). Further, if F1, ..., Fd are continuous,then C is unique, and given byC(u1, ..., ud) = F(F−11 (u1), ..., F−1d (ud)) for all (u1, ..., ud) ∈ [0, 1]d7
  • Arthur CHARPENTIER - Archimedean copulas.Definition 6. Let ψ be an generator of order d, i.e. ψ is decreasing andψ(1) = 0, the inverse ψ−1is d − 2 times continuously differentiable on(0, ∞), and (ψ−1)(d−2)is convex. ThenC(u1, ..., ud) = ψ←(ψ(u1) + ... + ψ(ud)), u1, ..., ud ∈ [0, 1],is a copula, called an Archimedean copula, with generator ψ.Note that ψ is a generator in any dimension d if and only if ψ(1) = 0 andψ−1is completely monotone, i.e. (−1)k(ψ−1)(k)(·) ≥ 0 on (0, ∞).8
  • Arthur CHARPENTIER - Archimedean copulas.• the lower Fréchet bound, ψ(t) = 1 − t, C−(u, v) = min{u + v − 1, 0},• the independent copula, ψ(t) = − log t, C⊥(u, v) = uv,• the Clayton’s copula, ψ(t) = t−θ− 1, C(u, v) = (uθ+ vθ− 1)−1/θ,• the Gumbel’s copula, ψ(t) = (− log t)−θ,C(u, v) = exp − (− log u)θ+ (− log v)θ 1/θ,• the Nelsen’s copula, ψ(t) = (1 − t)/t, C(u, v) = uv/(u + v − uv),0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0The lower Fréchet bound0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0The independent copula0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0Gumbel’s copula0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0Clayton’s copula0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0Nelsen’s copula9
  • Arthur CHARPENTIER - Archimedean copulas.Some more examples of Archimedean copulasψ(t) range θ(1) 1θ(t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)(2) (1 − t)θ [1, ∞)(3) log1−θ(1−t)t[−1, 1) Ali-Mikhail-Haq(4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986)(5) − log e−θt−1e−θ−1(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987)(6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993)(7) − log{θt + (1 − θ)} (0, 1](8) 1−t1+(θ−1)t[1, ∞)(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)(10) log(2t−θ − 1) (0, 1](11) log(2 − tθ) (0, 1/2](12) ( 1t− 1)θ [1, ∞)(13) (1 − log t)θ − 1 (0, ∞)(14) (t−1/θ − 1)θ [1, ∞)(15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994)(16) ( θt+ 1)(1 − t) [0, ∞)Table 1: Archimedean copulas, from Nelsen (2006).10
  • Arthur CHARPENTIER - Archimedean copulas.Where do these copulas come from ?• The conditional independence and frailty approachConsider two risks, X and Y , such thatX|Θ = θG ∼ E(θG) and Y |Θ = θG ∼ E(θG) are independent,X|Θ = θB ∼ E(θB) and Y |Θ = θB ∼ E(θB) are independent,(unobservable good (G) and bad (B) risks).The following figures start from 2 classes of risks, then 3, and then acontinuous risk factor θ ∈ (0, ∞).11
  • Arthur CHARPENTIER - Archimedean copulas.0 5 10 1505101520Conditional independence, two classes−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, two classesFigure 1: Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).12
  • Arthur CHARPENTIER - Archimedean copulas.0 5 10 15 20 25 30010203040Conditional independence, three classes−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, three classesFigure 2: Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).13
  • Arthur CHARPENTIER - Archimedean copulas.0 20 40 60 80 100020406080100Conditional independence, continuous risk factor−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, continuous risk factorFigure 3: Continuous classes, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).14
  • Arthur CHARPENTIER - Archimedean copulas.0 20 40 60 80 100020406080100Conditional independence, continuous risk factor−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, continuous risk factorFigure 4: Continuous classes, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).15
  • Arthur CHARPENTIER - Archimedean copulas.Assume that, given Θ, X|Θ ∼ E(αΘ) and Y |Θ ∼ E(βΘ) are independent,P(X > x, Y > y) =∞0P(X > x, Y > y|Θ = θ)π(θ)dθ=∞0P(X > x|Θ = θ)P(Y > y|Θ = θ)π(θ)dθ=∞0exp(−αθx) exp(−βθy)π(θ)dθ=∞0[exp(−[αx + βy]θ)] π(θ)dθ,where ψ(t) = E(exp −tΘ) = exp(−tθ)π(θ)dθ is the Laplace transform of Θ.16
  • Arthur CHARPENTIER - Archimedean copulas.Hence P(X > x, Y > y) = φ(αx + βy). Or,P(X > x) =∞0P(X > x|Θ = θ)π(θ)dθ=∞0exp(−αθx)π(θ)dθ= φ(αx),and thus αx = φ−1(P(X > x)) (similarly for βy). And therefore,P(X > x, Y > y) = φ(φ−1(P(X > x)) + φ−1(P(Y > y)))= C(P(X > x), P(Y > y)),setting C(u, v) = φ(φ−1(u) + φ−1(v)) for any (u, v) ∈ [0, 1] × [0, 1].Using any Laplace transforms, one can generate several families ofmultivariate distributions.Example 7. If Θ is Gamma distributed, the associated copula is Clayton’s.If Θ has an α-stable distributed, the associated copula is Gumbel’s.17
  • Arthur CHARPENTIER - Archimedean copulas.This approach can be used in motor insurance ratemaking, and in credit risk.A finite sequence {X1, ..., Xd} of random variables is exchangeable, ord-exchangeable, if(X1, ..., Xd)L= Xσ(1), ..., Xσ(d) , (1)for any permutation σ of {1, ..., d}. More generally, an infinite sequence{X1, X2...} of random variables is infinitely exchangeable (or simplyexchangeable) if(X1, X2, ...)L= Xσ(1), Xσ(2), ... , (2)for any finite permutation σ of N∗(that is Card {i, σ (i) = i} < ∞).A d-exchangeable sequence {X1, ..., Xd} is called m-extendible (for somem > d), if (X1, ..., Xd)L= (Z1, ..., Zd), where {Z1, ..., Zm} is somem-exchangeable sequence.18
  • Arthur CHARPENTIER - Archimedean copulas.Using the formulation of Aldous (1985), de Finetti’s theorem states that“aninfinite exchangeable sequence is a mixture of i.i.d. sequences”: X1, X2, ... ofBernoulli random variables is exchangeable if and only there is a randomvariable Θ, taking values in [0, 1] such that, given Θ = θ the Xi’s areindependent, and Xi ∼ B(θ) (see Schervish (1995) or Chow & Teicher(1997)).Example 8. This result can be easily interpreted in credit risk, wherevariables of interest are dichotomous (default or non-default). Let X1, X2, ...be an infinite exchangeable sequence of Bernoulli variables, and letSn = X1 + .... + Xn the number of defaults within n companies (for a givenperiod of time). Then, the distribution of Sn is a mixture of binomialdistributions, i.e. there is a distribution function H on [0, 1] such thatP (Sn = k) =10nkωk(1 − ω)n−kdH (θ) .19
  • Arthur CHARPENTIER - Archimedean copulas.• The survival distribution approachAssume that for a random vector (X, Y ), there exists a convex survivaldistribution S, such that S(0) = 1 andP(X > x, Y > y) = S(x + y),then the joint survival copula of (X, Y ), such thatP(X > x, Y > y) = C(P(X > x), P(Y > y),is C(u, v) = S(S−1(u) + S−1(v)), which is an Archimedean copula withgenerator ψ = S−1.This is the notion of Schur-constant survival distribution of random pair(X, Y ).Example 9. If S is the survival Pareto distribution, the associated copula isClayton’s. If S is the survival Weibull distribution, the associated copula isGumbel’s.20
  • Arthur CHARPENTIER - Archimedean copulas.• The serial iterate approachA “natural” idea to define d dimension copula from 2 dimensional copulascan be the serial iterate approach. Given a 2-copula C2, define recursivelyCn(u1, . . . , un−1, un) = C2(Cn−1(u1, . . . , un−1), un).Proposition 10. C is an associative copula, i.e.C(u, C(v, w) = C(C(u, v), w) for all u, v, w ∈ [0, 1], such that C(u, u) < u forall u ∈ (0, 1) if and only if C is Archimedean.Hence, the only copulas that can be constructed by serial iteration areArchimedean copulas.Remark 11. This is actually where the word Archimedean comes from.21
  • Arthur CHARPENTIER - Archimedean copulas.• The distorted copula approachDefinition 12. A distortion function is a function h : [0, 1] → [0, 1] strictlyincreasing such that h(0) = 0 and h(1) = 1.The set of distortion function will be denoted H.Note that h ∈ H if and only if h−1∈ H. Given a copula C, defineCh(u, v) = h−1(C(h(u), h(v))).If h is convex, then Ch is a copula, called distorted copula.Example 13. if h(x) = x1/n, the distorted copula isCh(u, v) = Cn(u1/n, v1/n), for all n ∈ N, (u, v) ∈ [0, 1]2.if the survival copula of the (Xi, Yi)’s is C, then the survival copula of(Xn:n, Yn:n) = (max{X1, ..., Xn}, max{Y1, ..., Yn}) is Ch.Example 14. if C(u, v) = uv = C⊥(u, v) (the independent copula), andφ(·) = log h(·), thenCh(u, v) = h−1(h(u)h(v)) = φ−1(φ(u) + φ(v)).22
  • Arthur CHARPENTIER - Archimedean copulas.• Kendall’s distribution approachArchimedean copulas can also be characterized through Kendall’s cdf, K,K(t) = P(C(U, V ) ≤ t), t ∈ [0, 1].where (U, V ) has cdf C.Note that K(t) = t − λ(t) where λ(t) = ψ(t)/ψ (t). And conversely ψ isψ(u) = ψ(u0) expuu01λ(t)dt for all 0 < u0 < 1.23
  • Arthur CHARPENTIER - Archimedean copulas.• Characterizations of some archimedean copulas1. Frank copula is the only Archimedean such that (U, V )L= (1 − U, 1 − V )(stability by symmetry),2. Clayton copula is the only Archimedean such that (U, V ) has the samecopula as (U, V ) given (U ≤ u, V ≤ v), for all u, v ∈ (0, 1] (stability bytruncature),3. Gumbel copula is the only Archimedean such that (U, V ) has the samecopula as (max{U1, ..., Un}, max{V1, ..., Vn}) for all n ≥ 1 (max-stability),24
  • Arthur CHARPENTIER - Archimedean copulas.Copula density0.0 0.4 0.80.00.51.01.52.0Archimedean generator0 1 2 3 4 5 60.00.40.8Laplace TransformLevel curves of the copula0.0 0.4 0.8−0.4−0.20.0Lambda function0.0 0.4 0.80.00.40.8Kendall cdfFigure 5: (Independent) Archimedean copula (C = C⊥, ψ(t) = − log t).25
  • Arthur CHARPENTIER - Archimedean copulas.Some “famous” Archimedean copulasClayton’s copula (Figure 6), with parameter α ∈ [0, ∞) has generatorψ(x; α) =x−α− 1αif 0 < α < ∞, with the limiting case ψ(x; 0) = − log(x), for any 0 < x ≤ 1.The inverse function is the Laplace transform of a Gamma distribution.The associated copula isC(u, v; α) = (u−α+ v−α− 1)−1/αif 0 < α < ∞, with the limiting case C(u, v; 0) = C⊥(u, v), for any(u, v) ∈ (0, 1]2.26
  • Arthur CHARPENTIER - Archimedean copulas.Copula density0.0 0.4 0.80.00.51.01.52.0Archimedean generator0 1 2 3 4 5 60.00.20.40.60.81.0Laplace TransformLevel curves of the copula0.0 0.4 0.8−0.4−0.3−0.2−0.10.0Lambda function0.0 0.4 0.80.00.20.40.60.81.0Kendall cdfFigure 6: Clayton’s copula.27
  • Arthur CHARPENTIER - Archimedean copulas.Gumbel’s copula (Figure 7), with parameter α ∈ [1, ∞) has generatorψ(x; α) = (− log x)αif 1 ≤ α < ∞, with the limiting case ψ(x; 0) = − log(x), for any 0 < x ≤ 1.The inverse function is the Laplace transform of a 1/α-stable distribution.The associated copula isC(u, v; α) = −1αlog 1 +(e−αu− 1) (e−αv− 1)e−α − 1,if 1 ≤ α < ∞, for any (u, v) ∈ (0, 1]2.28
  • Arthur CHARPENTIER - Archimedean copulas.Copula density0.0 0.4 0.80.00.51.01.52.0Archimedean generator0 1 2 3 4 5 60.00.20.40.60.81.0Laplace TransformLevel curves of the copula0.0 0.4 0.8−0.4−0.3−0.2−0.10.0Lambda function0.0 0.4 0.80.00.20.40.60.81.0Kendall cdfFigure 7: Gumbel’s copula.29
  • Arthur CHARPENTIER - Archimedean copulas.How to define more general parametric families ?Given an Archimedean generator ψ, define ψα and ψβ as follows,ψα,1(x) = ψ(xα) and ψ1,β(x) = ψ(x)β,where β ≥ 1 and α ∈ (0, 1]. Note that a composite family can also beconsidered, ψα,β(x) = [ψ(xα)]β.30
  • Arthur CHARPENTIER - Archimedean copulas.Distorted Archimedean copula, from Frankβ = 1α = 1Distorted Archimedean copula, from Frankβ = 3 2α = 1Distorted Archimedean copula, from Frankβ = 1α = 1 2Distorted Archimedean copula, from Frankβ = 3 2α = 1 2Figure 8: Distorted Archimedean copula (φα,β), from Frank copula.31
  • Arthur CHARPENTIER - Archimedean copulas.A short word on the zero-set areaDefine the zero-set boundary curve φ(u) + φ(v) = φ(0). If φ(0) = ∞, thezero-set boundary curve has a null measure (e.g. Clayton’s copula). Ifφ(0) < ∞, zero-set boundary curve is −φ(0)/φ (0) (e.g. Copula 4.2.2 inNelsen (2006)).Example 15. Clayton’s copula can be defined for θ ∈ [−1, 0) ∪ (0, +∞), asC(u, v) = max{ u−θ+ v−θ− 1−1/θ, 0}.If θ < 0, this copula has a zero-set, below the curve y = (1 − x−θ)−1/θExample 16. Consider Copula 4.2.2 in Nelsen (2006), defined forθ ∈ [1, +∞), asC(u, v) = max{1 − (1 − u)θ+ (1 − v)θ 1/θ, 0},with generator (1 − t)θ.32
  • Arthur CHARPENTIER - Archimedean copulas.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Generating Clayton’s copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Generating Clayton’s copulaFigure 9: Clayton’s copula, θ = 2 and θ = −1/2.33
  • Arthur CHARPENTIER - Archimedean copulas.0.0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0Generating copula 4.2.20.0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0Generating copula 4.2.2Figure 10: Copula 4.2.2, θ = 2 and θ = 5.34
  • Arthur CHARPENTIER - Archimedean copulas.Sequences of Archimedean copulasExtension of results due to Genest & Rivest (1986),Proposition The five following statements are equivalent,(i) limn→∞Cn(u, v) = C(u, v) for all (u, v) ∈ [0, 1]2,(ii) limn→∞ψn(x)/ψn(y) = ψ(x)/ψ (y) for all x ∈ (0, 1] and y ∈ (0, 1) such thatψ such that is continuous in y,(iii) limn→∞λn(x) = λ(x) for all x ∈ (0, 1) such that λ is continuous in x,(iv) there exists positive constants κn such that limn→∞ κnψn(x) = ψ(x) forall x ∈ [0, 1],(v) limn→∞Kn(x) = K(x) for all x ∈ (0, 1) such that K is continuous in x.35
  • Arthur CHARPENTIER - Archimedean copulas.Proposition 17. The four following statements are equivalent(i) limn→∞Cn(u, v) = C+(u, v) = min(u, v) for all (u, v) ∈ [0, 1]2,(ii) limn→∞λn(x) = 0 for all x ∈ (0, 1),(iii) limn→∞ψn(y)/ψn(x) = 0 for all 0 ≤ x < y ≤ 1,(iv) limn→∞Kn(x) = x for all x ∈ (0, 1).Note that one can get non Archimedean limits,0.0 0.4 0.80510150.0 0.4 0.80.00.20.40.60.81.0Sequence of generators and Kendall cdf’s36
  • Arthur CHARPENTIER - Archimedean copulas.Statistical inference for Archimedean copulasRecall that the Archimedean generator can be expressed asψ(u) = ψ(u0) expuu01λ(t)dt for all 0 < u0 < 1,where λ(t) = t − K(t), K begin Kendall’s distribution function, i.e.K(t) = P(F(X, Y ) ≤ t), t ∈ [0, 1].where F(x, y) = P(X ≤ x, Y ≤ y).Note that K can be writtenK(t) = Pr[F(X, Y ) ≤ t] = E[1{F(X, Y ) ≤ t}]=∞0∞01[F(x, y) ≤ t]dF(x, y).37
  • Arthur CHARPENTIER - Archimedean copulas.Thus, a natural estimator can be defined asK(t) =∞0∞01[F(x, y) ≤ t]dF(x, y)where F is a nonparametric estimate of the joint cdf F.Hence, a natural estimate for φ is thenφ(u) = expuu01λ(t)dt = expuu01t − K(t)dtwhich leads to Cφ (see Genest & Rivest (1993)).38
  • Arthur CHARPENTIER - Archimedean copulas.−2 −1 0 1 2−2−1012Scatterplot0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Kendall’s distribution function0.0 0.2 0.4 0.6 0.8 1.0−0.25−0.20−0.15−0.10−0.050.00Lambda function0.0 0.2 0.4 0.6 0.8 1.00123456Archimedean generatorFigure 11: Estimation of the Archimedean generator, n = 100.39
  • Arthur CHARPENTIER - Archimedean copulas.−2 −1 0 1 2−2−1012Scatterplot0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Kendall’s distribution function0.0 0.2 0.4 0.6 0.8 1.0−0.25−0.20−0.15−0.10−0.050.00Lambda function0.0 0.2 0.4 0.6 0.8 1.001234567Archimedean generatorFigure 12: Estimation of the Archimedean generator, n = 100.40
  • Arthur CHARPENTIER - Archimedean copulas.Generating Archimedean copulasIn dimension 2, recall that if (U, V ) has joint distribution C.P (V ≤ v|U = u) = limh→0P (V ≤ v|u ≤ U < u + h)= limh→0P (U < u + h, V ≤ v) − P (U < u, V ≤ v)P (U < u + h) − P (U < u)= limh→0C(u + h, v) − C(u, v)(u + h) − u= limh→0C(u + h, v) − C(u, v)(u + h) − u=∂C∂u (u,v).The general algorithm is thenU ← Random, and V ←∂C(U, ·)∂u−1(Random),Genest & MacKay (1986), Genest (1987) and Lee (1993) proposed thefollowing algorithm to generate random vectors (X, Y ) with Archimedean41
  • Arthur CHARPENTIER - Archimedean copulas.copula.U ⇐= Random,V is the solution of Random =(φ−1) (V ))(φ−1) (0).Note that other algorithms can be used, to generate pairs (U, V )U ⇐=Random, T ⇐=Random,W ⇐= (φ )−1(φ (U)/T)V ⇐= φ−1(φ(W) − φ(U))Or equivalentlyW ⇐= K−1(Random), S ⇐=Random,U ⇐= φ−1(Sφ(W))V ⇐= φ−1((1 − S)φ(W))42
  • Arthur CHARPENTIER - Archimedean copulas.Histogram, first component0.0 0.2 0.4 0.6 0.8 1.0020406080100−2 0 2 4−3−2−1012Gaussian distribution N(0,1)GaussiandistributionN(0,1)0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Uniform distribution on [0,1]Uniformdistributionon[0,1]Histogram, second component0.0 0.2 0.4 0.6 0.8 1.0020406080100Figure 13: Simulations using Clayton’s copula.43
  • Arthur CHARPENTIER - Archimedean copulas.Tails for Archimedean copulas (Fisher-Tippett)Standard approach, introduce λL and λU (lower and upper tail indices),λL = limu→0P X ≤ F−1X (u) |Y ≤ F−1Y (u) ,λU = limu→1P X > F−1X (u) |Y > F−1Y (u) .Those measures are copula based, i.e. λL = limu→0C(u, u)uFor Archimedean copulas, note thatλU = 2 − limx→01 − φ−1(2x)1 − φ−1(x)and λL = limx→0φ−1(2φ(x))x= limx→∞φ−1(2x)φ−1(x).44
  • Arthur CHARPENTIER - Archimedean copulas.ψ(t) range θ λL λU(1) 1θ(t−θ − 1) [−1, 0) ∪ (0, ∞) max(2−1/θ, 0) 0(2) (1 − t)θ [1, ∞) 0 2 − 21/θ(3) log1−θ(1−t)t[−1, 1) 0 0(4) (− log t)θ [1, ∞) 0 2 − 21/θ(5) − log e−θt−1e−θ−1(−∞, 0) ∪ (0, ∞) 0 0(6) − log{1 − (1 − t)θ} [1, ∞) 0 2 − 21/θ(7) (θ − 1) log{θt + (1 − θ)} (0, 1] 0 0(8) 1−t1+(θ−1)t[1, ∞) 0 0(9) log(1 − θ log t) (0, 1] 0 0(10) log(2t−θ − 1) (0, 1] 0 0(11) log(2 − tθ) (0, 1/2] 0 0(12) ( 1t− 1)θ [1, ∞) 2−1/θ 2 − 21/θ(13) (1 − log t)θ − 1 (0, ∞) 0 0(14) (t−1/θ − 1)θ [1, ∞) 1/2 2 − 21/θ(15) (1 − t1/θ)θ [1, ∞) 0 2 − 21/θ(16) ( θt+ 1)(1 − t) [0, ∞) 1/2 0(17) − log(1+t)−θ−12−θ−1(−∞, 0) ∪ (0, ∞) 0 0(18) eθ/(t−1) [2, ∞) 0 1(19) eθ/t − eθ (0, ∞) 1 0(20) e−tθ− e (0, ∞) 1 0(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) 0 2 − 21/θ(22) arcsin(1 − tθ) (0, 1] 0 045
  • Arthur CHARPENTIER - Archimedean copulas.Truncature of Archimedean copulasDefinition 18. Let U = (U1, ..., Un) be a random vector with uniformmargins, and distribution function C. Let Cr denote the copula of randomvector(U1, ..., Un)|U1 ≤ r1, ..., Ud ≤ rd, (3)where r1, ..., rd ∈ (0, 1].If Fi|r(·) denotes the (marginal) distribution function of Ui given{U1 ≤ r1, ..., Ui ≤ ri, ..., Ud ≤ rd} = {U ≤ r},Fi|r(xi) =C(r1, ..., ri−1, xi, ri+1, ..., rd)C(r1, ..., ri−1, ri, ri+1, ..., rd),and therefore, the conditional copula (or truncated copula) isCr(u) =C(F←1|r(u1), ..., F←d|r(ud))C(r1, ..., rd). (4)46
  • Arthur CHARPENTIER - Archimedean copulas.Proposition 19. The class of Archimedean copulae is stable by truncature.More precisely, if U has cdf C, with generator ψ, U given {U ≤ r}, for anyr ∈ (0, 1]d, will also have an Archimedean generator, with generatorψr(t) = ψ(tc) − ψ(c) where c = C(r1, ..., rd).0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.0 Generators of conditional Archimedean copulae(1) (2)(3)47
  • Arthur CHARPENTIER - Archimedean copulas.Tails for Archimedean copulas (Pickands-Balkema-de Haan)Proposition 20. Let C be an Archimedean copula with generator ψ, and0 ≤ α ≤ ∞. If C(·, ·; α) denote Clayton’s copula with parameter α.(i) limu→0 Cu(x, y) = C(x, y; α) for all (x, y) ∈ [0, 1]2;(ii) −ψ ∈ R−α−1.(iii) ψ ∈ R−α.(iv) limu→0 uψ (u)/ψ(u) = −α.If α = 0 (tail independence),(i) ⇐⇒ (ii)=⇒(iii) ⇐⇒ (iv),and if α ∈ (0, ∞] (tail dependence),(i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (iv).48
  • Arthur CHARPENTIER - Archimedean copulas.Proposition 21. There exists Archimedean copulae, with generators havingcontinuous derivatives, slowly varying such that the conditional copula doesnot convergence to the independence.Generator ψ integration of a function piecewise linear, with knots 1/2k,If −ψ ∈ R−1, then ψ ∈ Πg (de Haan class), and not ψ /∈ R0.This generator is slowly varying, with the limiting copula is not C⊥.Note that lower tail index isλL = limu↓0C(u, u)u= 2−1/α,with proper interpretations for α equal to zero or infinity (see e.g. Theorem3.9 of Juri and Wüthrich (2003)).49
  • Arthur CHARPENTIER - Archimedean copulas.ψ(t) range θ α(1) 1θ(t−θ − 1) [−1, 0) ∪ (0, ∞) max(θ, 0)(2) (1 − t)θ [1, ∞) 0(3) log1−θ(1−t)t[−1, 1) 0(4) (− log t)θ [1, ∞) 0(5) − log e−θt−1e−θ−1(−∞, 0) ∪ (0, ∞) 0(6) − log{1 − (1 − t)θ} [1, ∞) 0(7) (θ − 1) log{θt + (1 − θ)} (0, 1] 0(8) 1−t1+(θ−1)t[1, ∞) 0(9) log(1 − θ log t) (0, 1] 0(10) log(2t−θ − 1) (0, 1] 0(11) log(2 − tθ) (0, 1/2] 0(12) ( 1t− 1)θ [1, ∞) θ(13) (1 − log t)θ − 1 (0, ∞) 0(14) (t−1/θ − 1)θ [1, ∞) 1(15) (1 − t1/θ)θ [1, ∞) 0(16) ( θt+ 1)(1 − t) [0, ∞) 1(17) − log(1+t)−θ−12−θ−1(−∞, 0) ∪ (0, ∞) 0(18) eθ/(t−1) [2, ∞) 0(19) eθ/t − eθ (0, ∞) ∞(20) e−tθ− e (0, ∞) 0(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) 0(22) arcsin(1 − tθ) (0, 1] 050
  • Arthur CHARPENTIER - Archimedean copulas.Tails for Archimedean copulas (Pickands-Balkema-de Haan)Analogy with lower tails.Recall that ψ(1) = 0, and therefore, using Taylor’s expansion yieldsψ(1 − s) = −sψ (1) + o(s) as s → 0.And moreover, since ψ is convex, if ψ(1 − ·) is regularly varying with indexα, then necessarily α ∈ [1, ∞). If if (−D)ψ(1) > 0, then α = 1 (but theconverse is not true).0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.7Archimedean copula at 10.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.6Archimedean copula at 10.5 0.6 0.7 0.8 0.9 1.00.000.020.040.060.080.100.12Archimedean copula at 10.5 0.6 0.7 0.8 0.9 1.0−0.020.000.020.040.060.080.10Archimedean copula at 151
  • Arthur CHARPENTIER - Archimedean copulas.Proposition 22. Let C be an Archimedean copula with generator ψ.Assume that f : s → ψ(1 − s) is regularly varying with index α ∈ [1, ∞) andthat −ψ (1) = κ. Then three cases can be considered(i) if α ∈ (1, ∞), case of asymptotic dependence,(ii) if α = 1 and if κ = 0, case of dependence in independence,(iii) if α = 1 and if κ > 0, case of independence in independence.52
  • Arthur CHARPENTIER - Archimedean copulas.ψ(t) range θ α κ(1) 1θ(t−θ − 1) [−1, 0) ∪ (0, ∞) 1 1(2) (1 − t)θ [1, ∞) θ 0(3) log1−θ(1−t)t[−1, 1) 1 1 − θ(4) (− log t)θ [1, ∞) θ 0(5) − log e−θt−1e−θ−1(−∞, 0) ∪ (0, ∞) 1 θe−θe−θ−1(6) − log{1 − (1 − t)θ} [1, ∞) θ 0(7) − log{θt + (1 − θ)} (0, 1] 1 θ(8) 1−t1+(θ−1)t[1, ∞) 1 1/θ(9) log(1 − θ log t) (0, 1] 1 θ(10) log(2t−θ − 1) (0, 1] 1 2θ(11) log(2 − tθ) (0, 1/2] 1 θ(12) ( 1t− 1)θ [1, ∞) θ 0(13) (1 − log t)θ − 1 (0, ∞) 1 θ(14) (t−1/θ − 1)θ [1, ∞) θ 0(15) (1 − t1/θ)θ [1, ∞) θ 0(16) ( θt+ 1)(1 − t) [0, ∞) 1 θ + 1(17) − log(1+t)−θ−12−θ−1(−∞, 0) ∪ (0, ∞) 1 −θ2−θ−12−θ−1(18) eθ/(t−1) [2, ∞) ∞ 0(19) eθ/t − eθ (0, ∞) 1 θeθ(20) e−tθ− e (0, ∞) 1 θe(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) θ 0(22) arcsin(1 − tθ) (0, 1] 1 θ(·) (1 − t) log(t − 1) 1 053
  • Arthur CHARPENTIER - Archimedean copulas.0.0 0.2 0.4 0.6 0.8 1.00246810Archimedean copula density on the diagonalDependenceDependence in independenceIndependence in independenceCopula density54
  • Arthur CHARPENTIER - Archimedean copulas.A short word on hierarchical copulasAs pointed out in Genest, Quesada Molina & Rodríguez Lallena(1995), or Li, Scarsini & Shaked (1996) it is usually difficult to definecopulas as follows,C(u1, u2, u3, u4) = C0(C1,2(u1, u2), C3,4(u3, u4))where only 2-dimensional copulas are considered. But this can be donesimply in the case of Archimedean copulas. Consider φ, ψ and λ threeArchimedean copulas, and setC(u1, u2, u3, u4) = λ←(λ ◦ ψ←(ψ(u1) + ψ(u2)) + λ ◦ φ←(φ(u3) + φ(u4))).Under weak conditions such a function defines a 4 dimensional copula.55