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  1. 1. Arthur CHARPENTIER - Dependence between extremal eventsDependence between extremal eventsArthur CharpentierHong Kong University, February 2007Seminar of the department of Statistics and Actuarial Science1
  2. 2. Arthur CHARPENTIER - Dependence between extremal events• Lower tail dependence for Archimedean copulas:characterizations and pitfalls, (2006), to appear, InsuranceMathematics and Economics, with J. Segers,(http://www.crest.fr/.../charpentier-segers-ime.pdf)• Limiting dependence structures for tail events, with applicationsto credit derivatives , (2006), Journal of Applied Probability, 43, 563 -586, with A. Juri, (http://projecteuclid.org/.../pdf)• Convergence of Archimedean Copulas, (2006), to appear, Probabilityand Statistical Letters, with J. Segers, (http://papers.ssrn.com/...900113)• Tails of Archimedean Copulas, (2006), submitted, with J. Segers,(http://www.crest.fr.../Charpentier-Segers-JMA.pdf)2
  3. 3. Arthur CHARPENTIER - Dependence between extremal events“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 extreme valuetheory in some way.” Mikosch (2005).3
  4. 4. Arthur CHARPENTIER - Dependence between extremal events“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 modelbuilding (and Archimedean copulas in particular) seems to have been largelyfuelled at the end of the 1980s by the publication of significant papers byMarshall and Olkin (1988) and by Oakes (1989) in the influential Journal ofthe American Statistical Association”.“The work of Pickands (1981) and Deheuvels (1982) also led several authorsto adhere to the copula point of view in studying multivariate extremes”.Genest & Rémillard (2006).4
  5. 5. Arthur CHARPENTIER - Dependence between extremal eventsDefinition 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 copulaFigure 1: Copula C(u, v) and its density c(u, v) = ∂2C(u, v)/∂u∂v.5
  6. 6. Arthur CHARPENTIER - Dependence between extremal eventsTheorem 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 ).6
  7. 7. Arthur CHARPENTIER - Dependence between extremal eventsNote that if (X, Y ) has copula C,P(X ≤ x, Y ≤ y) = C(P(X ≤ x), P(Y ≤ y))for all (u, v) ∈ [0, 1] × [0, 1], and equivalentlyP(X > x, Y > y) = C∗(P(X > x), P(Y > y))for all (u, v) ∈ [0, 1] × [0, 1].C∗is a copula, called the survival copula of pair (X, Y ), and it satisfiesC∗(u, v) = u + v − 1 + C(1 − u, 1 − v) for all (u, v) ∈ [0, 1] × [0, 1].Note that if (U, V ) has distribution C, then C∗is the distribution function of(1 − U, 1 − V ).7
  8. 8. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Scatterplot (U,V) from copula CFirst component, USecondcomponent,V0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Scatterplot (1−U,1−V) from survival copula C*First component, 1−USecondcomponent,1−V−3 −2 −1 0 1 2 3−3−2−10123Scatterplot (X,Y) from copula CFirst component, XSecondcomponent,Y−3 −2 −1 0 1 2 3−3−2−10123Scatterplot (−X,−Y) from survival copula C*First component, −XSecondcomponent,−YFigure 2: Scatterplot of C (pair U, V ) and C∗(pair 1 − U, 1 − V ).8
  9. 9. Arthur CHARPENTIER - Dependence between extremal eventsIn 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(u, v) = ψ←(ψ(u) + ψ(v)), u, v ∈ [0, 1],is a copula, called an Archimedean copula, with generator ψ.9
  10. 10. Arthur CHARPENTIER - Dependence between extremal events• 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,• Clayton’s copula, ψ(t) = t−θ− 1, C(u, v) = (uθ+ vθ− 1)−1/θ,• Gumbel’s copula, ψ(t) = (− log t)−θ,C(u, v) = exp − (− log u)θ+ (− log v)θ 1/θ,• 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 copula10
  11. 11. Arthur CHARPENTIER - Dependence between extremal eventsP(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=∞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.Hence P(X > x, Y > y) = φ(αx + βy). Similarly,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].11
  12. 12. Arthur CHARPENTIER - Dependence between extremal events0 5 10 1505101520Conditional independence, two classes−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, two classesFigure 3: Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).12
  13. 13. Arthur CHARPENTIER - Dependence between extremal events0 5 10 15 20 25 30010203040Conditional independence, three classes−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, three classesFigure 4: Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).13
  14. 14. Arthur CHARPENTIER - Dependence between extremal events0 20 40 60 80 100020406080100Conditional independence, continuous risk factor−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, continuous risk factorFigure 5: Continuous classes, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).14
  15. 15. Arthur CHARPENTIER - Dependence between extremal events0 20 40 60 80 100020406080100Conditional independence, continuous risk factor−3 −2 −1 0 1 2 3−3−2−10123Conditional independence, continuous risk factorFigure 6: Continuous classes, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).15
  16. 16. Arthur CHARPENTIER - Dependence between extremal eventsCopula 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 7: (Independent) Archimedean copula (C = C⊥, ψ(t) = − log t).16
  17. 17. Arthur CHARPENTIER - Dependence between extremal eventsClayton’s copula (Figure 8), 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]2.17
  18. 18. Arthur CHARPENTIER - Dependence between extremal eventsCopula 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 8: Clayton’s copula.18
  19. 19. Arthur CHARPENTIER - Dependence between extremal eventsGumbel’s copula (Figure 9), 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]2.19
  20. 20. Arthur CHARPENTIER - Dependence between extremal eventsCopula 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 9: Gumbel’s copula.20
  21. 21. Arthur CHARPENTIER - Dependence between extremal eventsModeling joint extremal events“The extension of univariate results is not entirely immediate : the obviousproblem is the lack of natural order in higher dimension.” (Tawn (1988)).Consider (Xi) an i.i.d. sequence of random variables, with commondistribution function FX . Define, for all n ∈ N∗the associated statistic order(Xi:n) and Xn the average, i.e.X1:n ≤ X2:n ≤ ... ≤ Xn:n and Xn =X1 + ... + Xnn.Assume that V ar (X) < ∞, from the central limit theorem, if an = E (X) andbn = V ar (X) /n,limn→∞PXn − anbn≤ x = Φ (x) ,where Φ denotes the c.d.f. of the standard normal distribution.21
  22. 22. Arthur CHARPENTIER - Dependence between extremal eventsMore generally, if X /∈ L2or X /∈ L1, analogous results could be obtainedAssume thatlimn→∞PXn − anbn≤ x = G (x) .The set of nondegenerate function is the set of stable distributions, a subset ofinfinitely divisible distributions (see Feller (1971) or Petrov (1995)).The limiting distributions can be characterized through their Laplacetransform.22
  23. 23. Arthur CHARPENTIER - Dependence between extremal eventsIn the case of the maxima, consider an i.i.d. sequence of random variables,X1, X2, ..., with common distribution function FX, F(x) = P{Xi ≤ x}. ThenP{Xn:n ≤ x} = FX (x)n.This result simply says that for any fixed x for which F(x) < 1,P{Xn:n ≤ x} → 0. Hence,Xn:nP−as→ xF = sup{x ∈ R, FX(x) < 1},and if X is not bounded Xn:nP−as→ xF = ∞23
  24. 24. Arthur CHARPENTIER - Dependence between extremal eventsIn order to obtain some asymptotic distribution for Xn:n, one should consideran affine transformation, i.e. find an > 0, bn such thatPXn:n − bnan≤ x = F(anx + bn)n→ H(x),for some nondegenerated function H.The limiting distibution necessarily satisfies some stability condition, i.e.H(anx + bn)n= H(x) for some an > 0, bn, for any n ∈ N. Hence, H satisfiesthe following functional equationH(a(t)x + b(t))t= H(x) for all x, t ≥ 0.24
  25. 25. Arthur CHARPENTIER - Dependence between extremal eventsThe so-called Fisher-Tippett theorem (see Fisher and Tippett (1928),Gnedenko (1943)), asserts that if a nondegenerate H exists (i.e. adistribution function which does not put all its mass at a single point), itmust be one of three types:• H (x) = exp (−x−γ) if x > 0, α > 0, the Fréchet distribution,• H (x) = exp (− exp (−x)), the Gumbel distribution,• H (x) = exp − (−x)−γif x < 0 ,α > 0 , the Weibull distribution.25
  26. 26. Arthur CHARPENTIER - Dependence between extremal eventsThe three types may be combined into a single Generalised Extreme Value(GEV) distribution:Hξ,µ,σ(x) = exp − 1 + ξx − µσ−1/ξ+, (2.6)(where y+ = max(y, 0)) where µ is a location parameter, σ > 0 is a scaleparameter and ξ is a shape parameter.• the limit ξ → 0 corresponds to the Gumbel distribution,• ξ > 0 to the Fréchet distribution with γ = 1/ξ,• ξ < 0 to the Weibull distribution with γ = −1/ξ.26
  27. 27. Arthur CHARPENTIER - Dependence between extremal eventsFurthermore, note that• µ and σ depend on the affine transformation, an and bn,• ξ depends on the distribution F.Definition 4. If there are an and bn such that a non-degenerate limit exists,FX will be said to be in the max-domain of attraction of Hξ, denotedFX ∈ MDA (Hξ).The exponential and the Gaussian distributions have light tails (ξ = 0), andthe Pareto distribution has heavy tails (ξ > 0).27
  28. 28. Arthur CHARPENTIER - Dependence between extremal eventsIn order to characterize distributions in some max-domain of attraction, let usintroduce the following concept of regular variation.Definition 5. A measurable function f : (0, ∞) → (0, ∞) is said to beregularly varying with index α at infinity, denoted f ∈ Rα (∞) iflimu→∞f (ux)f (u)= xα.If α = 0, the function will be said to be slowly varying. Notice that f ∈ Rα ifand only if there is L slowly varying such that f (x) = xαL (x).Proposition 6. If FX ∈ Rα (∞), α < 0, then the limiting distribution isFréchet with index −α, i.e. H−1/α. Analogous properties could be obtained ifξ ≤ 0 .28
  29. 29. Arthur CHARPENTIER - Dependence between extremal eventsConsider the distribution of X conditionally on exceeding some high thresholdu,Fu(y) = P{X − u ≤ y | X > u} =F(u + y) − F(u)1 − F(u).As u → xF = sup{x : F(x) < 1}, we often find a limitFu(y) ∼ G(y; σu, ξ),where G is Generalised Pareto Distribution (GPD) defined asG(y; σ, ξ) = 1 − 1 + ξyσ−1/ξ+. (2.8)The Gaussian distribution has light tails (ξ = 0). The associated limitingdistribution is the exponential distribution.29
  30. 30. Arthur CHARPENTIER - Dependence between extremal eventsTheorem 7. For ξ ∈ R, the following assertions are equivalent,1. F ∈ MDA (Hξ), i.e. there are (an) and (bn) such thatlimn→∞P (Xn:n ≤ anx + bn) = Hξ (x) , x ∈ R.2. There exists a positive, measurable function a (·) such that for 1 + ξx > 0,limu→∞F (u + xa (u))F (u)= limu→∞PX − ua (u)> x |X > u=(1 + ξx)−1/ξif ξ = 0,exp (−x) if ξ = 0.30
  31. 31. Arthur CHARPENTIER - Dependence between extremal eventsThe general structure for such bivariate extreme value distributions has beenknown since the end of the 50’s, due to Tiago de Olivera (1958),Geoffroy (1958) or Sibuya (1960). Those three papers obtained equivalentrepresentations (in dimension 2 or higher).Most of the results on multivariate extremes have been obtained consideringcomponentwise ordering, i.e. considering possible limiting distributions for(Xn:n, Yn:n). As pointed out in Tawn (1988) “A difficulty with this approachis that in some applications it may be impossible for (Xn:n, Yn:n) to occur as avector observation”. Despite this problem, this is the approach most widelyused in bivariate extreme value analysis.31
  32. 32. Arthur CHARPENTIER - Dependence between extremal events−4 −2 0 2 4−3−2−10123Maximum componentwiseFirst componentSecondcomponent−4 −2 0 2 4−3−2−10123Joint exceedance approachFirst componentSecondcomponentFigure 10: Modeling joint extremal events.32
  33. 33. Arthur CHARPENTIER - Dependence between extremal eventsSuppose that there are sequences of normalizing constant αX,n, αY,n > 0 andβX,n, βY,n such thatPXn:n − βX,nαX,n≤ x,Yn:n − βY,nαY,n≤ y= FnX,Y (αX,nx + βX,n, αY,ny + βY,n) → G (x, y) ,as n → ∞, where G is a proper distribution function, non-degenerated in eachmargin.Bivariate extreme value distributions are obtained as limiting distributions oflimn→∞PXn:n − anbn≤ x,Yn:n − cndn≤ y = C (HξX (x) , HξY (y)) .i.e. the normalized distribution of the vector of componentwise maxima.33
  34. 34. Arthur CHARPENTIER - Dependence between extremal eventsC is called an extreme value copula,C (u, v) = exp (log u + log v) Alog ulog u + log v, (1)where 0 < u, v < 1, and A is a convex function on [0, 1] such thatA+(t) = max {t, 1 − t} ≤ A (t) ≤ 1 = A⊥(t).(see Capéraà, Fougères and Genest (1997), based on Pickands (1981)).Example 8. If A(ω) = exp (1 − ω)θ+ ωθ 1/θ, then C is Gumbel copula.Further, if A (ω) = max {1 − αω, 1 − β (1 − ω)}, where 0 ≤ α, β ≤ 1, then C isMarshall and Olkin copula.34
  35. 35. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00.50.60.70.80.91.0Pickands dependence function A0.0 0.2 0.4 0.6 0.8 1.00.50.60.70.80.91.0Pickands dependence function AFigure 11: Gumbel, and Marshall & Olkin’s dependence function A(ω).35
  36. 36. Arthur CHARPENTIER - Dependence between extremal eventsProposition 9. Consider (X1, Y1), ..., (Xn, Yn), ... sequence of i.i.d. versionsof (X, Y ), with c.d.f. (X, Y ). Assume that there are normalizing sequencesαX,n, αY,n, αX,n, αY,n > 0 and βX,n, βY,n, βX,n, βY,n such thatFnX,Y (αX,nx + βX,n, αY,ny + βY,n) → G (x, y)FnX,Y αX,nx + βX,n, αY,ny + βY,n → G (x, y) ,as n → ∞, for two non-degenerated distributions G and G . Then marginaldistributions of G and G are unique up to an affine transformation, i.e. thereare αX, αY , βX, βY such thatGX (x) = GX (αXx + βX) and GY (y) = GY (αY y + βY ).Further, the dependence structures of G and G are equal, i.e. the copulae areequal, CG = CG .Frank copula has independence in tails (A = A⊥) and the survival Claytoncopula has dependence in tails (A = A⊥). The associated limited copula isGumbel.36
  37. 37. Arthur CHARPENTIER - Dependence between extremal eventsJoe (1993) defined, in the bivariate case a tail dependence measure.Definition 10. Let (X, Y ) denote a random pair, the upper and lower taildependence parameters are defined, if the limit exist, asλL = limu→0P X ≤ F−1X (u) |Y ≤ F−1Y (u) ,andλU = limu→1P X > F−1X (u) |Y > F−1Y (u) .Proposition 11. Let (X, Y ) denote a random pair with copula C, the upperand lower tail dependence parameters are defined, if the limit exist, asλL = limu→0C(u, u)uand λU = limu→1C∗(u, u)1 − u.Example 12. If (X, Y ) has a Gaussian copula with parameter θ < 1, thenλ = 0.37
  38. 38. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Marges uniformesCopuledeGumbel−2 0 2 4−2024Marges gaussiennesFigure 12: Simulations of Gumbel’s copula θ = 1.2.38
  39. 39. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Marges uniformesCopuleGaussienne−2 0 2 4−2024Marges gaussiennesFigure 13: Simulations of the Gaussian copula (θ = 0.95).39
  40. 40. Arthur CHARPENTIER - Dependence between extremal eventsExample 13. Consider the case of Archimedean copulas, thenλU = 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).Ledford and Tawn (1996) propose the following model to study taildependence. Consider standardized marginal variables, with unit Fréchetdistributions, such thatP(X > t, Y > t) ∼ L(t) · [P(X > t)]1/η, t → ∞,where L denotes some slowly varying functions, and η ∈ (0, 1] will be calledcoefficient of tail dependence,• η describes the kind of limiting dependence,• L describes the relative strength, given η.40
  41. 41. Arthur CHARPENTIER - Dependence between extremal eventsMore precisely,• η = 1, perfect positive dependence (tail comontonicity),• 1/2 < η < 1 and L → c > 0, more dependent than independence, butasymptotically independent,• η = 1/2, tail independence• 0 < η < 1/2 less dependent than independence.Example 14. : distribution with Gumbel copula,P(X ≤ x, Y ≤ y) = exp(−(x−α+ y−α)1/α), α ≥ 0then η = 1 and (t) → (2 − 21/θ).41
  42. 42. Arthur CHARPENTIER - Dependence between extremal eventsA short word on tail parameter estimationFor the estimation of η, defineT =11 − FX (X)∧11 − FY (Y ),then FT , is regularly varying with parameter η: Hill’s estimator can be used.42
  43. 43. Arthur CHARPENTIER - Dependence between extremal eventsunivariate case bivariate caselimiting distribution dependence structure ofof Xn:n (G.E.V.) componentwise maximumwhen n → ∞ (Xn:n, Yn:n)(Fisher-Tippet)limiting distribution dependence structure ofof X|X > x (G.P.D.) (X, Y ) |X > x, Y > ywhen x → ∞ when x, y → ∞(Balkema-de Haan-Pickands)43
  44. 44. Arthur CHARPENTIER - Dependence between extremal eventsConditional copulaeLet U = (U1, ..., Un) be a random vector with uniform margins, anddistribution function C. Let Cr denote the copula of random vector(U1, ..., Un)|U1 ≤ r1, ..., Ud ≤ rd, (2)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 isCr(u) =C(F←1|r(u1), ..., F←d|r(ud))C(r1, ..., rd). (3)44
  45. 45. Arthur CHARPENTIER - Dependence between extremal eventsA bivariate regular variation propertyIn the univariate case, h is regularly varying if therelimt→0h(tx)h(t)= λ(x), for all x > 0.For all x, y > 0, limt→0h(txy)h(t)= λ(xy), andlimt→0h(txy)h(t)= limt→0h(txy)h(tx)×h(tx)h(t)= λ(y) × λ(x).Thus, necessarily λ(xy) = λ(x) × λ(y). It is Cauchy functional equation andthus, necessarily, λ(x) = xθ(power function) for some θ ∈ R.45
  46. 46. Arthur CHARPENTIER - Dependence between extremal eventsIn the (standard) bivariate case (see Resnick (1981)), h is regularly varying iftherelimt→0h(tx, ty)g(t, t)= λ(x, y), for all x, y > 0.This will be called ray-convergence. Then, there is θ ∈ R such thatλ(tx, ty) = tθλ(x, y) (homogeneous function).46
  47. 47. Arthur CHARPENTIER - Dependence between extremal eventsA general extention is to consider (see Meerschaert & Scheffer (2001)) isto assume that there is a sequence (At) of operators, regularly varying withindex E such thatlimt→0hA−1txy · g(t)−1= λ(x, y).Then there is θ ∈ R such that λ(tExy) = tθλ(x, y) (generalizedhomogeneous function).47
  48. 48. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Bivariate regular variation, ray convergenceFirst component, XSecondcomponent,Y0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Bivariate regular variation, directional convergenceFirst component, XSecondcomponent,YFigure 14: Two concepts of regular variation in R2.48
  49. 49. Arthur CHARPENTIER - Dependence between extremal eventsA subdefinition has been proposed by de Haan, Omey & Resnick (1984), adirectional convergence: given r, s : [0, 1] → [0, 1] such that r(t), s(t) → 0 ast → 0, both regularly varying (with index α and β respectively), thenlimt→0h(r(t)x, s(t)y) · g(t)−1= λ(x, y),then there is θ ∈ R such thatλ(tαx, tβy) = tθλ(x, y),for all x, y, t > 0, i.e. λ is a (generalized homogeneous function).49
  50. 50. Arthur CHARPENTIER - Dependence between extremal eventsLet C denote a copula, such that C(u, v) > 0 for all u, v > 0. Furthermore,consider r and s two continuous functions, regularly varying at 0, r ∈ Rα ands ∈ Rβ , so that s(t), r(t) → 0 when t → 0, so thatlimt→0C(r(t)x, s(t)yC(r(t), s(t))= φ(x, y), (4)where φ is a positive measurable function.Then φ satisfies the following functional equation φ(tα, tβ) = tθφ(x, y) forsome θ > 0. Hence, φ is a so-called generalized homogeneous function (seeAczél (1966)), which has an explicit general solution (in dimension 2). tThemost general solution is given byφ(x, y) =xθ/αh(yx−β/α) if x = 0cyθ/βif x = 0 and y = 00 if x = y = 0, (5)where c is a constant and h is function of one variable.50
  51. 51. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Marshall and Olkin’s copula Level curves of the copulaDISCONTINUITYMarshall and Olkin’s copulaFigure 15: Marshall and Olkin’s copula.51
  52. 52. Arthur CHARPENTIER - Dependence between extremal events2 4 6 8 10 12 144681012Scatterplot, LOSS−ALAELosses amounts (log)Allocatedexpenses(log)0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Copula density, LOSS−ALAEFigure 16: On statistical inference for tail events.52
  53. 53. Arthur CHARPENTIER - Dependence between extremal events2 4 6 8 10 12 144681012Scatterplot, LOSS−ALAE, maximum componentwiseLosses amounts (log)Allocatedexpenses(log)0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Copula density, LOSS−ALAE, n=502 4 6 8 10 12 144681012Scatterplot, LOSS−ALAE, maximum componentwiseLosses amounts (log)Allocatedexpenses(log)0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Copula density, LOSS−ALAE, n=50Figure 17: On statistical inference for tail events.53
  54. 54. Arthur CHARPENTIER - Dependence between extremal events2 4 6 8 10 12 144681012Scatterplot, LOSS−ALAE, joint−exceedencesLosses amounts (log)Allocatedexpenses(log)0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Copula density, LOSS−ALAE, u=85%2 4 6 8 10 12 144681012Scatterplot, LOSS−ALAE, joint−exceedencesLosses amounts (log)Allocatedexpenses(log)0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Copula density, LOSS−ALAE, u=95%Figure 18: On statistical inference for tail events.54
  55. 55. Arthur CHARPENTIER - Dependence between extremal eventsConditional dependence for Archimedean copulaeProposition 15. 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 1.00.00.51.01.52.02.53.0Generators of conditional Archimedean copulae(1) (2)(3)55
  56. 56. Arthur CHARPENTIER - Dependence between extremal eventsArchimedean copulae in lower tailsProposition 16. 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).56
  57. 57. Arthur CHARPENTIER - Dependence between extremal eventsProposition 17. 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)).Frank copula has independence in tails (C = C⊥) and the 4-2-14 copula hasdependence in tails (C = C⊥). The associated limited copula is Clayton.57
  58. 58. Arthur CHARPENTIER - Dependence between extremal eventsψ(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] 058
  59. 59. Arthur CHARPENTIER - Dependence between extremal eventsArchimedean copulae in upper tailsAnalogy 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 index α,then necessarily α ∈ [1, ∞). If if (−D)ψ(1) > 0, then α = 1 (but the converseis 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 159
  60. 60. Arthur CHARPENTIER - Dependence between extremal eventsProposition 18. Let C be an Archimedean copula with generator ψ. Assumethat f : s → ψ(1 − s) is regularly varying with index α ∈ [1, ∞) and that−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.60
  61. 61. Arthur CHARPENTIER - Dependence between extremal eventsψ(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) 1 061
  62. 62. Arthur CHARPENTIER - Dependence between extremal events0.0 0.2 0.4 0.6 0.8 1.00246810Archimedean copula density on the diagonalDependenceDependence in independenceIndependence in independenceCopula density62
  63. 63. Arthur CHARPENTIER - Dependence between extremal eventsOn sequences of Archimedean copulaeExtension 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.63
  64. 64. Arthur CHARPENTIER - Dependence between extremal eventsProposition 19. 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’s64

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