On tail dependence coecients of transformed multivariate Archimedean copulas

Transcription

1 Tails and for Archim Copula () February 2015, University of Lille 3 On tail dependence coecients of transformed multivariate Archimedean copulas Elena Di Bernardino, CNAM, Paris, Département IMATH Séminaire de statistique et économétrie February 2015, University of Lille 3

2 Tails and for Archim Copula () February 2015, University of Lille 3 Contents Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration 4 Denition Transformed Archimedean copulas with given tail coecients 5

3 Tails and for Archim Copula () February 2015, University of Lille 3 (Multivariate) Return Period The notion of Return Period (RP) is frequently used in environmental sciences for the identication of dangerous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the Value-at-Risk in Economics and Finance, since it is used to quantify and assess the risk. During the last years, researchers in environmental elds joined eorts to properly answer the following crucial question: How is it possible to calculate the critical design event(s) in the multivariate case?. In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al.(2011), Salvadori et al.(2012), Gräler et al. (2013) Multivariate return period using the notion of upper and lower level sets of multivariate probability distribution F and of the associated Kendall's measure.

4 Tails and for Archim Copula () February 2015, University of Lille 3 (Multivariate) Return Period The notion of Return Period (RP) is frequently used in environmental sciences for the identication of dangerous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the Value-at-Risk in Economics and Finance, since it is used to quantify and assess the risk. During the last years, researchers in environmental elds joined eorts to properly answer the following crucial question: How is it possible to calculate the critical design event(s) in the multivariate case?. In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al.(2011), Salvadori et al.(2012), Gräler et al. (2013) Multivariate return period using the notion of upper and lower level sets of multivariate probability distribution F and of the associated Kendall's measure.

5 ails and for Archim Copula () February 2015, University of Lille 3 (Multivariate) Return Period The notion of Return Period (RP) is frequently used in environmental sciences for the identication of dangerous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the Value-at-Risk in Economics and Finance, since it is used to quantify and assess the risk. During the last years, researchers in environmental elds joined eorts to properly answer the following crucial question: How is it possible to calculate the critical design event(s) in the multivariate case?. In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al.(2011), Salvadori et al.(2012), Gräler et al. (2013) Multivariate return period using the notion of upper and lower level sets of multivariate probability distribution F and of the associated Kendall's measure.

6 ails and for Archim Copula () February 2015, University of Lille 3 (Multivariate) Return Period The notion of Return Period (RP) is frequently used in environmental sciences for the identication of dangerous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the Value-at-Risk in Economics and Finance, since it is used to quantify and assess the risk. During the last years, researchers in environmental elds joined eorts to properly answer the following crucial question: How is it possible to calculate the critical design event(s) in the multivariate case?. In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al.(2011), Salvadori et al.(2012), Gräler et al. (2013) Multivariate return period using the notion of upper and lower level sets of multivariate probability distribution F and of the associated Kendall's measure.

7 ails and for Archim Copula () February 2015, University of Lille 3 (Multivariate) Return Period The notion of Return Period (RP) is frequently used in environmental sciences for the identication of dangerous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the Value-at-Risk in Economics and Finance, since it is used to quantify and assess the risk. During the last years, researchers in environmental elds joined eorts to properly answer the following crucial question: How is it possible to calculate the critical design event(s) in the multivariate case?. In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al.(2011), Salvadori et al.(2012), Gräler et al. (2013) Multivariate return period using the notion of upper and lower level sets of multivariate probability distribution F and of the associated Kendall's measure.

8 Tails and for Archim Copula () February 2015, University of Lille 3 Critical layers Consider the (nonnegative) real-valued random vector X = (x 1,..., X d ) such that X FX = C(F X1,..., F Xd ), with FX : R d + [0, 1]. Denition (Critical layer) The critical layer L(α) associated to the multivariate distribution function FX of level α (0, 1) is dened as L(α) = {x R d + : FX(x) = α}. Then L(α) is the iso-hyper-surface (with dimension d 1) where F equals the constant value α. The critical layer L(α) partitions R d into three non-overlapping and exhaustive regions: L < (α) = {x R d : FX(x) < α}, L(α) = the critical layer itself, L > (α) = {x R d : FX(x) > α}.

9 Tails and for Archim Copula () February 2015, University of Lille 3 Multivariate RP and Critical layers Event of interest is of the type {X A}, where A is a non-empty Borel set in R d collecting all the values judged to be dangerous according to some suitable criterion. A natural choice for A is the set L > (α) Then RP > (α) = t, where P[X L > t > 0 is the (α)] (deterministic) average time elapsing between X k and X k+1, k IN. Then, the considered Return Period can be expressed using Kendall's function RP > (α) = t 1 1 K C (α), where K C (α) = P [ X L < (α) ] = P [C(U 1,..., U d ) α], for α (0, 1).

10 Tails and for Archim Copula () February 2015, University of Lille 3 Goals and ideas This talk aims at: Giving a parametric representation of the multivariate distribution F of a d random vector X of risks using transformations Giving direct estimation procedure for this representation Giving closed parametric expressions, both for critical layers L(α) and Return Periods RP > (α) Working on the tails by using generators exhibiting any chosen couple of lower and upper tail dependence coecients

11 Tails and for Archim Copula () February 2015, University of Lille 3 Goals and ideas This talk aims at: Giving a parametric representation of the multivariate distribution F of a d random vector X of risks using transformations Giving direct estimation procedure for this representation Giving closed parametric expressions, both for critical layers L(α) and Return Periods RP > (α) Working on the tails by using generators exhibiting any chosen couple of lower and upper tail dependence coecients

12 Tails and for Archim Copula () February 2015, University of Lille 3 Goals and ideas This talk aims at: Giving a parametric representation of the multivariate distribution F of a d random vector X of risks using transformations Giving direct estimation procedure for this representation Giving closed parametric expressions, both for critical layers L(α) and Return Periods RP > (α) Working on the tails by using generators exhibiting any chosen couple of lower and upper tail dependence coecients

13 Tails and for Archim Copula () February 2015, University of Lille 3 Goals and ideas This talk aims at: Giving a parametric representation of the multivariate distribution F of a d random vector X of risks using transformations Giving direct estimation procedure for this representation Giving closed parametric expressions, both for critical layers L(α) and Return Periods RP > (α) Working on the tails by using generators exhibiting any chosen couple of lower and upper tail dependence coecients

14 Tails and for Archim Copula () February 2015, University of Lille 3 Goals and ideas This talk aims at: Giving a parametric representation of the multivariate distribution F of a d random vector X of risks using transformations Giving direct estimation procedure for this representation Giving closed parametric expressions, both for critical layers L(α) and Return Periods RP > (α) Working on the tails by using generators exhibiting any chosen couple of lower and upper tail dependence coecients

15 Part I. of Archimedean copulas

16 Tails and for Archim Copula () February 2015, University of Lille 3 Copulas Let F be a d dimensional cdf with marginals F i, i = 1,..., d. By Sklar's theorem, there exists a copula function C : [0, 1] d [0, 1] that links the distribution F with its margins: F (x 1,..., x d ) = C(F 1(x 1),..., F d (x d )). C is a d dimensional cdf on [0, 1] d with uniform marginals. C is unique if marginals F i are continuous.

17 Tails and for Archim Copula () February 2015, University of Lille 3 Archimedean copulas Archimedean copulas: C(u 1,..., u d ) = φ ( φ 1 (u 1) φ 1 (u d ) ) φ : R + (0, 1] is the generator of the Archimedean copula, φ 1 (x) = inf { s R + : φ(s) x } its (generalized) inverse function. Generator: φ is continuous, decreasing, d monotone (cf. McNeil and Ne²lehová, 2009), φ(0) = 1, φ(x) = 0. lim x + One limitation: we consider only here strict generators, i.e. x R +, φ(x) > 0. φ is strictly decreasing and φ 1 is the regular inverse of φ. Proposition (Equivalent generators, cf. Nelsen) Generator φ a(x) = φ(ax) and φ(x) lead to the same copula, a R \ {0} implies that one can ask φ to be such that φ(t 0) = ϕ 0 for an arbitrary point (t 0, ϕ 0) R (0, 1).

18 Tails and for Archim Copula () February 2015, University of Lille 3 Transformed Archimedean copulas Proposition (Transformed copula) Copula C depends on a continuous increasing function T : [0, 1] [0, 1], C(u 1,..., u d ) = T (C 0(T 1 (u 1),..., T 1 (u d ))). and if the initial C 0 is a given Archimedean copula with generator φ 0, then C is an Archimedean copula with generator φ(x) = T φ 0(x) Admissibility conditions for φ (d-monotonicity, cf. McNeil and Ne²lehová, 2009) more admissibility conditions for T (e.g. using Faa Di Bruno formula). Literature on these transformations: Durrleman et al. (2000), Charpentier (2008), Valdez and Xiao (2011).

19 Tails and for Archim Copula () February 2015, University of Lille 3 Motivations Possibility to transform both copulas and margins: F (x 1,..., x d ) = C( F 1(x 1),..., F d (x d )), where F i = T T 1 i F i and T i : [0, 1] [0, 1] are continuous and increasing. Why using transformations (instead of parametric multivariate distributions)? 1 Flexible: allows transformations composition huge variety of reachable distributions (multimodal, etc.) possibility to improve a t gradually 2 Invertible: analytical expressions for the expression of the distribution function F ; but also for the expression of the critical layer L(α) 3 facilities

20 Part II. of Transformed Archimedean copulas Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration

21 Tails and for Archim Copula () February 2015, University of Lille 3 Diagonal of the copula Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Idea: building a non-parametric estimator of the generator φ and for the transformation T based on the diagonal section of the copula. Denition (Diagonal section of the copula) Consider a copula C satisfying regular conditions. For all u [0, 1], δ 1(u) = C(u,..., u), and δ 1 is the inverse function of δ 1 so that δ 1 δ 1 = Id. Remark: The copula C is not always uniquely determined by its diagonal (cf. Frank's condition, Erdely et al. 2013) based only on this diagonal may fail to capture tail dependence when φ (0) = (upper tail dependent case).

22 Tails and for Archim Copula () February 2015, University of Lille 3 Denitions Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Denition (Discrete self-nested diagonals) Consider a copula C satisfying regular conditions. The discrete self-nested diagonal of C at order k is the function δ k such that for all u [0, 1], k IN δ k (u) = δ 1... δ 1(u), (k times), δ k (u) = δ 1... δ 1(u), (k times), δ 0(u) = u, where δ 1(u) = C(u,..., u) and δ 1 is the inverse function of δ 1, so that δ 1 δ 1 is the identity function. Denition (Self-nested diagonals) Functions of a family {δ r } r R are called (extended) self-nested diagonals of a copula C, if δ k (u) is the discrete self-nested diagonal of C at order k, for all k Z, and if furthermore δ r1 +r2(u) = δ r1 δ r2 (u), r 1, r 2 R, u [0, 1].

23 Self-nested diagonals Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Proposition (Self-nested diagonals of an Archimedean copula) If C is an Archimedean copula associated with a generator φ, then the self-nested diagonals of C at order r is δ r (x) = φ(d r φ 1 (x)), r R. - If C is the Independence copula of generator φ(t) = exp( t), then δ r (u) = u (dr ). - If C is a Gumbel copula of generator φ(t) = exp( t 1/θ ), then δ r (u) = u (d(r/θ) ), θ 1. - If C is a Clayton copula of generator φ(t) = (1 + θt) 1/θ, δ r (u) = (1 + d r (t θ 1)) 1/θ, θ > 0. Tails and for Archim Copula () February 2015, University of Lille 3

24 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration using self-nested diagonals Proposition (Transformation T using self-nested diagonals) Consider Archimedean copulas C 0 and C satisfying regular conditions and the associated self-nested diagonals δ r and δ r, r R. If T is dened by T(0) = 0, T (1) = 1 and for all x (0, 1), T (x) = δ r(x) (y 0), with r(x) such that δ r(x) (x 0) = x, then the transformed copula using transformation T is equal to C : for all u 1,..., u d, C(u 1,..., u d ) = T C 0(T 1 (u 1),..., T 1 (u d )), where (x 0, y 0) (0, 1) 2 can be arbitrarily chosen. In the case where C 0 is the independence copula, ( ) r(x) = 1 ln ln x. ln d ln x0

25 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Generators using self-nested diagonals Proposition (Generator φ using self-nested diagonals) Consider an Archimedean copula C satisfying regular conditions, and the associated self-nested copulas δ r, for r R. Assume that the copula C is reachable by transforming an Archimedean copula C 0, and denote by δ r, r R, the self-nested diagonals of C 0 and by φ 0 its generator. A generator φ of C is dened for all t R + by φ(t) = δ ρ(t) (y 0), with ρ(t) such that δ ρ(t) (x 0) = φ 0(t) i.e., ( ) ρ(t) = 1 ln t, ln d φ 1 0 (x 0) where (x 0, y 0) (0, 1) 2 can be arbitrarily chosen. In the particular case where C 0 is the independent copula, then ρ(t) = 1 ( ) t ln d ln ln x 0 Tails and for Archim Copula () February 2015, University of Lille 3

26 Part II. of the transformed Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration

27 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Non-parametric estimators of self-nested diagonals First, build a (smooth) estimator δ of the diagonal of the target transformed copula C (e.g. empirical copula, Deheuvels (1979), Fermanian et al. (2004), Omelka et al. (2009)), and denote its inverse function δ 1. Estimators of discrete self-nested diagonal of C at order k are the function δ k such that for all u [0, 1], k IN δ k (u) = δ 1... δ 1(u), (k times), δ k (u) = δ 1... δ 1(u), (k times), δ 0(u) = u,

28 Tails and for Archim Copula () February 2015, University of Lille 3 Non-parametric estimators of φ Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Denition (Non-parametric estimation of φ - Case C 0 independent copula) Consider an Archimedean copula C and associated self-nested diagonals δ r, for r R. Denote by δ r the estimator of δ r, for r R. Assume that φ(t 0) = ϕ 0, for a given couple of values (t 0, ϕ 0) R + \ {0} (0, 1). A non-parametric estimator φ of φ is dened by φ(0) = 1 and for all t R + \ {0}, φ(t) = δ ρ(t) (ϕ 0), ( ) with ρ(t) = 1 ln t, ln d where (t 0, ϕ 0) R + \ {0} (0, 1) can be arbitrarily chosen. In particular, the estimator φ of φ is passing through the points t0 {(t k, ϕ k )} k Z = {(d k t 0, δ k (ϕ 0))} k Z

29 Non-parametric φ(t) Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Figure : Estimated versus theoretical Gumbel-generator with parameter θ = 3. Size of simulated samples n = 150 (left) and n = 1500 (right). Estimated φ(t) = δ ρ(t) (y0) (full line). The theoretical standardized Gumbel-generator, i.e., φ(t) = exp( t 1/θ ), is drawn using a dashed line. We force the generators to pass through the point (t0, ϕ 0) = (1, e 1 ) (black point). Tails and for Archim Copula () February 2015, University of Lille 3

30 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Comparison with estimator of Genest et al. (2011) Recall the λ function, as originally introduced in Genest and Rivest (1993) for inferential purposes, λ(u) = φ 1 (u) φ (φ 1 (u)). One can easily see that this coecient is identical for any generator belonging to the same equivalent class and on the contrary of the generator itself, does not depend on the choice of some arbitrarily point (t 0, ϕ 0). Following the same methodology as Genest and Rivest (1993), we estimate the λ function, for our estimator and for the estimator of Genest et al. (2011). For our estimator, we propose: ˆλ(u) = ˆφ 1 (u) ˆφ( ˆφ 1 (u)+h) ˆφ( ˆφ 1 (u) h) 2h, for a small value of h, u (h, 1 h). ˆλ ˆδ (u) = 1 h (u) ˆδ h (u), ln d 2h (since λ(u) = ln 1 d r δr (u) r=0; this second simple estimator of λ permit to avoid function inversions φ 1 ).

31 Black: theoretical λ function. Dark green dashed line: estimator by Genest et al. (2011), Black dotted line: ˆλ(u). Violet dotted-dashed line: ˆλ (u).

32 Black: theoretical λ function. Dark green dashed line: estimator by Genest et al. (2011), Black dotted line: ˆλ(u). Violet dotted-dashed line: ˆλ (u).

33 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Comparison with estimator of Genest et al. (2011) The violet dashed line ˆλ (u) seems performing a little bit better On tested data, no estimator seems to perform signicantly better. However, in our case, estimators relying on self-nested diagonals have several advantages among which: The facility to get generators passing through a given point, contrary to the estimator in Genest et al. (2011) which relies on the choice of a radius r m. For instance, in this gures we had to nd of the Genest et al.'s estimator the value r m = 5500 to get a correct result. Both estimators of T or φ are relying on direct analytical expressions, whereas the estimator in Genest et al. (2011) rely on a large number of root resolution procedures. Indeed in the last estimator we have to solve a triangular non-linear system containing m equations. If the sample size is n = 2000 the value m is approximately around Some rst theoretical results on condence bands (see slides below). Such results would probably be dicult to get with estimators relying on successive optimization procedures or root resolutions. Tails and for Archim Copula () February 2015, University of Lille 3

34 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Non-parametric estimators of T Denition (Non-parametric estimation of T - Case C 0 independent copula) Consider an Archimedean copula C and associated self-nested diagonals δ r, for r R. Denote by δ r the estimator of δ r, for r R. A non-parametric estimator of T is dened by T (0) = 0, T (1) = 1 and for all x (0, 1) by T (x) = δ r(x) (y 0), ( with r(x) = 1 ln ln d ln x ln x0 where (x 0, y 0) (0, 1) 2 can be arbitrarily chosen. ),

35 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Non-parametric estimators of T and φ Same kind of estimator for T or φ in the general case: T (x) = δ r(x) (y 0), with r(x) such that δ r(x) (x 0) = x, where (x 0, y 0) (0, 1) 2 can be arbitrarily chosen. Proposition (Theoretical condence bands for estimators - not detailed here) Theoretical condence bands on δ theoretical condence bands on φ. one needs the distribution of the empirical process δ(u), u [0, 1]. Tails and for Archim Copula () February 2015, University of Lille 3

36 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Condence bands for δ 1 imply condence bands for φ Figure : (Left) Condence bands for δ 1 and δ 1 for chosen parameters α = β = 1.05, α + = β + = 0.9. (Right) Resulting theoretical condence band for φ. Here C is a Gumbel copula of parameter θ = 2, the size of generated sample is n = Tails and for Archim Copula () February 2015, University of Lille 3

37 Part II. of the transformed Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration

38 A class of parametric transformation We take back from Bienvenüe and Rullière (2012): Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Denition (Conversion and transformation functions) Let f any bijective increasing function from R to R. It is said to be a conversion function. The transformation T f : [0, 1] [0, 1] is dened as 0 if u = 0, T f (u) = logit 1 (f (logit(u))) if 0 < u < 1, 1 if u = 1. Remark: transformations function are chosen in a way to be easily invertible (T f T g = T f g, T 1 f = T f 1). We will use (composited) hyperbolic conversion functions: ( x m h f (x) = H m,h,p1,p2,η(x) = m h + (ep 1 + e p 2 ) x m h 2 (e p 1 e p 2 ) with m, h, p1, p2 R, and one smoothing parameter η R. f 1 (x) = H 1 m,h,p1,p2,η (x) = H m, h, p1, p2,η(x). G. Tails and for Archim Copula () February 2015, University of Lille 3 2 ) 2 + e η p 1 +p 2 2

39 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Parametric transformations estimation Idea: Transform an initial given copula C 0 using a transformation T, Transform initial given margins using transformations T 1,..., T d. : Get non-parametric estimators for T Get non-parametric estimators for T 1,..., T d Fit all transformations T, T 1,..., T d by piecewise-linear functions Approach piecewise linear functions by composition of hyperbolas H m,h,p1,p2,η, cf. Bienvenüe and Rullière (2012).

40 Part II. of the transformed Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration

41 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Geyser data: transformed (parametric) bivariate density Non-parametric Parametric estimation (without optimization). Data : 272 eruptions of the Old Faithful geyser in Yellowstone National Park. Each observation consists of two measurements: the duration (in min) of the eruption (X ), and the waiting time (in min) before the next eruption (Y ). Figure : Level curves of transformed density f (x1, x 2 ) and Old Faithful geyser data (red points). Left: parameter setting η = 0.9, η1 = 9, η 2 = 7.5; (right) parameter setting η = 0.9, η 1 = 4, η 2 = 4.

42 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Geyser data: transformed (parametric) c.d.f. F and L(α) Figure : (Left) Transformed distribution F (x1, x 2 ) with associated transformed level curves (red curves). (Right) Black points are plotted at empirical tail probabilities on the diagonal, calculated from the empirical bivariate distribution (in log scale). Red line is 1 F (x, x) (in log scale). The vertical dotted lines show estimate of 95% VaR (i.e., the univariate quantile in log scale). Tails and for Archim Copula () February 2015, University of Lille 3

43 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Geyser data: transformed (parametric) marginal F 1 Univariate parametric fit F_1 Tail probabilities for F_1 Fn(x) log(1 F_1) x log(x) Figure : Duration (in min) of the eruption data. (Left) F1 (red) and the empirical distribution function of eruption data (black). (Right) Black points are plotted at empirical tail probabilities calculated from empirical distribution function (in log scale). Red line is 1 F1 (x) (in log scale). The vertical dotted lines show estimate of 95% VaR (univariate quantile) for the eruption data.

44 Tails and for Archim Copula () February 2015, University of Lille 3 Self-nested diagonals Non-parametric estimation Parametric estimation Real-data illustration Geyser data: transformed (parametric) marginal F 2 Univariate parametric fit F_2 Tail probabilities for F_2 Fn(x) log(1 F_2) x log(x) Figure : Black points are plotted at empirical tail probabilities calculated from the empirical marginal distributions (in log scale). Red line is 1 F1 (x) (in log scale) (left) and 1 F2 (x) (in log scale) (right). The vertical dotted lines show estimate of 95% VaR (i.e., the univariate quantiles in log scale).

45 Part III. Denitions Transformed Archimedean copulas given tail coecients

46 Tails and for Archim Copula () February 2015, University of Lille 3 Tail problem Denition Transformed Archimedean copulas with given tail coecients Need for some applications (hydrology, nance, etc.). Need to know joint extreme behavior of some random variables Need to dene and capture tail dependence Limits of our non-parametric estimator: Here estimation relying on the diagonal section δ 1, The copula C is not always uniquely determined by its diagonal (cf. Frank's condition, Erdely et al. 2013) based only on this diagonal may fail to capture tail dependence when φ (0) =. Not usual to capture tail behavior for non-parametric estimation (e.g. empirical cdf, our generator estimator, Genest et al., 2011 one)

47 Tails and for Archim Copula () February 2015, University of Lille 3 Denition Transformed Archimedean copulas with given tail coecients Possible parametric solutions: here the transformed generator is φ = T φ 0 to reach a given tail dependence, possibility to work on the tails of the transformation T on the initial generator φ 0 to be transformed e.g. φ 0 with good tail dependence, T not changing tail dependence, e.g. φ 0 with correct central t, T changing only tails, etc. We work with the lower and upper tail dependence coecients (TDC). Denition (Bivariate tail dependence coecients - Sibuya (1960)) For d = 2, the classical bivariate upper and lower tail dependence coecients, λ U and λ L, are dened as λ L = lim u 0 + P[V u U u] and λ U = lim u 1 P[V > u U > u].

48 Tails and for Archim Copula () February 2015, University of Lille 3 Multivariate TDC Denition Transformed Archimedean copulas with given tail coecients Denition (Multivariate tail dependence coecients: TDC) Assume that the considered copula C is the distribution of some random vector U := (U 1,..., U d ). Denote I = {1,..., d} and consider two non-empty subsets I h I and Ī h = I \ I h of respective cardinal h 1 and d h 1. A multivariate version of classical bivariate tail dependence coecients is given by Li (2009) and De Luca and Rivieccio (2012) (when limits exist): λ I h,īh L = lim u 0 + P [ U i u, i I h U i u, i Ī h ], λ I h,īh U = lim u 1 P [ U i u, i I h U i u, i Ī h ]. If for all I h I, λ I h,īh L = 0, (resp. λ I h,īh U = 0) then we say U is lower tail independent (resp. upper tail independent).

49 Tails and for Archim Copula () February 2015, University of Lille 3 Denition Transformed Archimedean copulas with given tail coecients With exchangeable r.v., depends on d = card(i ) and h = card(i h ). In particular case of Archimedean copulas, one have (ψ = φ 1 ): Proposition (Multivariate Tail Dep. Coes. for Archimedean copulas) For Archimedean copulas the multivariate lower and upper tail dependence coecients are respectively: λ (h,d h) L = lim u 0 + λ (h,d h) U = lim u 1 ψ 1 (d ψ(u)) ψ 1 ((d h) ψ(u)), d ( 1)i C i i=0 d ψ 1 (i ψ(u)) d h i=0 ( 1)i C i d h ψ 1 (i ψ(u)). Some important references on tail dependence and Archimedean copulas: Juri and Wüthrich (2003), Charpentier and Segers (2009), Durante et al. (2010).

50 Denition Transformed Archimedean copulas with given tail coecients TDC for some usual copulas Clatyton copulas Gumbel copulas u Figure : Concentration function λ LU (u) = 1 {u 1/2}λ L (u) + 1 {u>1/2}λ U (u) for some Clayton copulas (left panel) and Gumbel copulas (right panel). u Tails and for Archim Copula () February 2015, University of Lille 3

51 Tails and for Archim Copula () February 2015, University of Lille 3 TDC for some usual copulas Denition Transformed Archimedean copulas with given tail coecients As noticed in the bivariate case by Avérous and Dortet-Bernadet (2004), many of the most commonly used parametric families of Archimedean copulas (such as the Clayton, Gumbel, Frank or Ali-Mikhail-Haq systems) possess strong dependence properties: they have the SI or the SD property and are ordered at least by <LTD. where Stochastic Increasingness (SI), Stochastically Decreasingness (SD) and Left-Tail Decreasingness (LTD) denitions are recalled in Joe (1997). It implies in particular that for many classical Archimedean copulas (and in particular for Clayton, Gumbel, Frank or Ali-Mikhail-Haq copulas) λ LU is nondecreasing on (0, 1/2) and non-increasing on (1/2, 1).

52 Tails and for Archim Copula () February 2015, University of Lille 3 Regular Variation (RV) Denition Transformed Archimedean copulas with given tail coecients At innity: f (s x) f RV α( ) s > 0, lim x + f (x) = sα. At zero, using M(x) = 1 x and I (x) = 1/x, At one, f RV α(0) f I RV α( ) s > 0, f RV α(1) f M I RV α( ) s > 0, f (s x) lim x 0 + f (x) = sα. f (1 s x) lim x 0 + f (1 x) = sα.

53 Part III. Denitions Transformed Archimedean copulas given tail coecients

54 Tails and for Archim Copula () February 2015, University of Lille 3 Considered transformations Denition Transformed Archimedean copulas with given tail coecients We consider transformations T f : [0, 1] [0, 1] such that 0 if u = 0, T f (u) = G f G 1 (u) if 0 < u < 1, 1 if u = 1, Why? The transformation T f has support [0, 1]. The function G is a cdf that aims at transferring this support on the whole real line R. The conversion function f is any continuous bijective increasing function, f : R R, without bounding constraints. cf Bienvenüe and Rullière.

55 Tails and for Archim Copula () February 2015, University of Lille 3 Denition Transformed Archimedean copulas with given tail coecients Considered transformations - Archimedean case Assumption (Considered transformed generators) Consider an initial Archimedean copula C 0 with generator φ 0, and the associated transformed one, C, with generator φ = T f φ 0, i.e. on (0, ), φ = G f G 1 φ 0. One assumes that generators φ 0 and φ satisfy admissibility conditions. The transformed generator will depend on the RV properties of f, φ 0 and G.

56 Tails and for Archim Copula () February 2015, University of Lille 3 Lower tail assumptions Denition Transformed Archimedean copulas with given tail coecients Assumption (Lower-tails: assumptions on f, φ 0, G) Assume that f, φ 0 and G are continuous and dierentiable functions, strictly monotone with respective proper inverse functions denoted f 1, ψ 0 = φ 1 and 0 G 1. Furthermore, i) The function f has a left asymptote f (x) = a x + b as x tends to, for a (0, + ) and b (, + ). ii) The inverse initial generator ψ 0 is regularly varying at 0 with some index r 0, that is ψ 0 RV r0 (0), with r 0 [0, + ]. iii) The function G is a non-defective continuous c.d.f. with support R. The following rate of G is regularly varying with some index g 1: m G = G /G RV g 1( ), with g (0, + ).

57 Tails and for Archim Copula () February 2015, University of Lille 3 Upper tail assumptions Denition Transformed Archimedean copulas with given tail coecients Assumption (Upper-tails: assumptions on f, φ 0, G) Assume that f, φ 0 and G are continuous and dierentiable functions, strictly monotone with respective proper inverse functions denoted f 1, ψ 0 = φ 1 and 0 G 1. Furthermore, i) The function f has a right asymptote f (x) = α x + β as x tends to +, for α (0, + ) and β (, + ). ii) The inverse initial generator ψ 0 is regularly varying at 1 with some index ρ 0, i.e., ψ 0 RV ρ0 (1), with ρ 0 [1, + ]. iii) The function G is a non-defective continuous c.d.f. with support R. The hazard rate of G is regularly varying with some index γ 1, that is µ G = G /Ḡ RV γ 1( ), with Ḡ = 1 G and γ (0, + ).

58 Tails and for Archim Copula () February 2015, University of Lille 3 TDC of transformed generators Denition Transformed Archimedean copulas with given tail coecients Theorem (Multivariate lower TDC of transformed Archimedean copula) Under Lower tail assumptions, i) the inverse transformed generator ψ is such that ψ RV r (0), with r = r 0 a g ii) the associated transformed lower multivariate lower tail dependence coecient is given by: see next slide, if r = 0, λ (h,d h) L = d ag r0 1 (d h) ag r0 1, if r (0, + ), 1, if r = +.

59 Tails and for Archim Copula () February 2015, University of Lille 3 TDC of transformed generators Denition Transformed Archimedean copulas with given tail coecients Theorem (Multivariate upper TDC of transformed Archimedean copula) Under Lower tail assumptions, i) the inverse transformed generator ψ is such that ψ RV ρ(1), with ρ = ρ 0 α γ and ρ [1, + ] ii) the associated transformed upper multivariate lower tail dependence coecient is given by: λ (h,d h) U = see next slide, if ρ = 1, di =1 C i d ( 1)i i αγ ρ0 1 d h i =1 C i d h ( 1)i i αγ ρ0 1, if ρ (1, + ), 1, if ρ = +.

60 Tails and for Archim Copula () February 2015, University of Lille 3 Denition Transformed Archimedean copulas with given tail coecients Asymptotic lower tail independence When ψ 0 RV r0 (0), with r 0 = 0, λ (h,d h) L = lim u 0 + λ(h,d h) L (u) = 0 if µ φ0 = φ 0 /φ0 RV k0 1( ), with k0 [0, + ), if T f RVã(0) for ã (0, + ), (h,d h) λ L (u) RV z (0) with z = d k0 (d h) k0. Asymptotic upper tail independence When ψ RV ρ(1), with ρ = 1 and if φ is a d times continuously dierentiable generator. λ (h,d h) U = lim u 1 λ(h,d h) U (u) = 0. if ( D) d φ(0) is nite and not zero, where D is the derivative operator, (h,d h) λ U (u) RV h (1); if ψ (1) = 0 and the function L(s) := s d ds { ψ(1 s) s } is positive and L RV 0(0), (h,d h) λ U (u) RV 0(1).

61 Denition Transformed Archimedean copulas with given tail coecients Transformed case with ρ = 1 Remarks: λ (h,d h) U (u) RV h (1); This case represents a upper asymptotic independence for C in a rather strong sense. It is called near independence. λ (h,d h) U (u) RV 0(1) This case represents a boundary between asymptotic independence and asymptotic dependence -case called near asymptotic dependence. Tails and for Archim Copula () February 2015, University of Lille 3

62 Part III. Denitions Transformed Archimedean copulas given tail coecients

63 Tails and for Archim Copula () February 2015, University of Lille 3 Fit with given TDC Denition Transformed Archimedean copulas with given tail coecients In the bivariate case, when d = 2, λ (1,1) L = 2 ag r0 1 and λ(1,1) U = 2 2 αγ ρ0 1 Take a transformation T f (x) = G f G 1 (x) with f = H m, h, p1, p2, η, H m, h, p1, p2, η(x) = m h+(ep1 +e p2 ) x m h (e p1 e p2 ) 2 ( x m h 2 ) 2 + e η p 1 +p 2 2, So, if these tail coecients are given, then we can easily nd a = e p1 and α = e p2 (1,1) (1,1) as functions of λ L and λ U : ( ) ( ) p 1 = 1 (1,1) g ln ln λ L r 0 and p 2 = 1 (1,1) ln 2 γ ln ln(2 λ U ) ρ 0. ln 2 In the multivariate case, when d > 2, same principle, but expression of p 2 more dicult to write.

64 Tails and for Archim Copula () February 2015, University of Lille 3 Illustration with logit transformations Denition Transformed Archimedean copulas with given tail coecients Let the initial copula C 0 be a Clayton copula with parameter θ > 0. Assume that we want to obtain an arbitrarily chosen couple of target tail (1,1) (1,1) coecients: λ L = 1/4 and λ U = 3/4. Let G = logit 1, then g = γ = 1. We obtain the parameters p 1 and p 2 of the hyperbolic conversion function H. chosen parameters deduced parameters tails coecients m h η θ p 1 p 2 λ (1,1) 0,L λ (1,1) 0,U λ (1,1) L λ (1,1) U A B C D

65 Transformed copulas with chosen TDC Denition Transformed Archimedean copulas with given tail coecients Models A, B, C, D u Figure : Tail concentration function λ LU (u) = 1 {u 1/2}λ L (u) + 1 {u>1/2}λ U (u) for some transformed copulas. Chosen values: λ L = 1/4 and λ U = 3/4. Full line corresponds to parameters A in Table before; dashed line to B; dotted line to C; dashed-dotted line to D. Tails and for Archim Copula () February 2015, University of Lille 3

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67 Tails and for Archim Copula () February 2015, University of Lille 3 On the non-parametric estimation of the generator: Comparable performance with other estimators, Without solving non-linear systems of equations, Theoretical condence bands. On the parametric estimation of the generator: Analytical expressions for the level curves, Tunable number of parameters, can be chosen Perspectives Estimators using other information (φ θ by classical estimation methods, e.g. tau's Kendall + transformed tails) Using nested copulas (multimodal distribution or non-exchangeable vectors - nested copula...) Comparisons of our multivariate extreme measures with existing multivariate risk measures.

68 Papers (partly) presented More details on Non-parametric estimation part: Di Bernardino, E. and Rullière, D. (2013). On certain transformations of Archimedean copulas : Application to the non-parametric estimation of their generators. Dependence Modeling 1(1): More details on Parametric estimation part: Di Bernardino, E. and Rullière, D. (2013). Distortions of multivariate distribution functions and associated level curves: Applications in multivariate risk theory. Insurance: Mathematics and Economics, 53(1): Di Bernardino, E. and Rullière, D. (2015). of multivariate critical layers: Applications to rainfall data. Journal de la Société Française de Statistique, to appear. More details on part: Di Bernardino, E. and Rullière, D. (2014). On tail dependence coecients of transformed multivariate Archimedean copulas. Preprint available on HAL. Tails and for Archim Copula () February 2015, University of Lille 3

69 Tails and for Archim Copula () February 2015, University of Lille 3 Thank you for your attention