Non parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015.

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1 Non parametric estimation of Archimedean copulas and tail dependence Elena Di Bernardino a and Didier Rullière b Paris, february 19, a CNAM, Paris, Département IMATH, b ISFA, Université Lyon 1, Laboratoire SAF. CNAM seminar, february /35

2 Part I. Transformations of Archimedean copulas

3 Copulas Transformations of Archimedean 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. CNAM seminar, february /35

4 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 Nešlehová, 2009), φ(0) = 1, lim φ(x) = 0. 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). CNAM seminar, february /35

5 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 Nešlehová, 2009) more admissibility conditions for T (e.g. using Faa Di Bruno formula). Literature on these transformations: Durrleman and al. (2000), Charpentier (2008), Valdez and Xiao (2011) CNAM seminar, february /35

6 Motivations Possibility to transform both copulas and margins: where F i = T T 1 i F (x 1,..., x d ) = C( F 1(x 1),..., F d (x d )), F i and T i : [0, 1] [0, 1] are continuous and increasing. Why using transformations (instead of parametric multivariate distributions)? 1 Flexible: allows distortions composition huge variety of reachable distributions (multimodal, etc.) possibility to improve a fit gradually 2 Invertible: analytical expressions for the expression of the distribution function; but also for the expression of level curves; 3 Estimation facilities CNAM seminar, february /35

7 Part II. Estimation of the distorted Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation

8 Diagonal of the copula Self-nested diagonals Non-parametric estimation Parametric estimation Idea: building a non-parametric estimator of the generator φ based on the diagonal section of the copula. Definition (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 = I d. Remark: The copula C is not always uniquely determined by its diagonal (cf. Frank s condition, Erdely and al. 2013) Estimation based only on this diagonal may fail to capture tail dependence when φ (0) =. CNAM seminar, february /35

9 Definitions Transformations of Archimedean copulas Self-nested diagonals Non-parametric estimation Parametric estimation Definition (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. Definition (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 +r 2 (u) = δ r1 δ r2 (u), r 1, r 2 R, u [0, 1]. CNAM seminar, february /35

10 Self-nested diagonals Self-nested diagonals Non-parametric estimation Parametric estimation Proposition (Self-nested diagonals of an Archimedean copula) If C is an Archimedean copula associated with a generator φ, then the self-nested diagonal of C at order r is δ r (x) = φ(d r φ 1 (x)), r R. Proposition (Interpolation of self-nested diagonals) Let C be an Archimedean copula with generator φ. For any real r [k, k + 1], k Z, the self-nested diagonals of C satisfies: δ r (x) = φ ( (φ 1 δ k (x) ) 1 α ( φ 1 δ k+1 (x) ) α ), x [0, 1], with α = r r and k = r, where r denotes the integer part of r. Interpolation does not depend on the Gumbel parameter in the Gumbel case. CNAM seminar, february /35

11 Self-nested diagonals Non-parametric estimation Parametric estimation Expression of distortions using self-nested diagonals Proposition (Distortion 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 defined 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 distorted copula using distortion 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 x ln d ln, ln x 0 CNAM seminar, february /35

12 Self-nested diagonals Non-parametric estimation Parametric estimation Expression of 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 distorting 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 defined 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), 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 CNAM seminar, february /35

13 Part II. Estimation of the distorted Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation

14 Self-nested diagonals Non-parametric estimation Parametric estimation Non-parametric estimators of self-nested diagonals I First, build a (smooth) estimator δ 1 of the diagonal of the target distorted copula C (e.g. empirical copula, Deheuvels (1979), Fermanian and al. (2004), Omelka and 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, CNAM seminar, february /35

15 Self-nested diagonals Non-parametric estimation Parametric estimation Non-parametric estimators of self-nested diagonals II Define self-nested diagonals at non-integer orders with a given interpolation function z, ( ( δ r (x) = z z 1 δ ) 1 α ( k (x) z 1 δ ) ) α k+1 (x), x [0, 1], with α = r r and k = r, where r denotes the integer part of r. Perfect interpolation functions z are functions such that for all r 1, r 2 R, δ r1 δ r2 = δ r1 +r 2. They do not depend on the Gumbel parameter in the Gumbel case (and in particular in Independent case). CNAM seminar, february /35

16 Non-parametric estimators of φ Self-nested diagonals Non-parametric estimation Parametric estimation Definition (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 defined 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 t 0 ) 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 {(t k, ϕ k )} k Z = {(d k t 0, δ k (ϕ 0))} k Z, No interpolation function z is needed to get (t k, ϕ k ), for k Z., CNAM seminar, february /35

17 Self-nested diagonals Non-parametric estimation Parametric estimation 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 confidence bands for estimators - not detailed here) Theoretical confidence bands on δ 1 theoretical confidence bands on φ. one needs the distribution of the empirical process δ(u), u [0, 1]. CNAM seminar, february /35

18 Non-parametric φ(t) Self-nested diagonals Non-parametric estimation Parametric estimation 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) (y 0) (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 (t 0, ϕ 0) = (1, e 1 ) (black point). CNAM seminar, february /35

19 Comparison with other estimators Self-nested diagonals Non-parametric estimation Parametric estimation lambda tau = 0.75, n=1000, d= Estimation of λ function, λ = φ 1 (φ φ 1 ). xval Black: theoretical one. Dark green dashed line: estimator proposed by Genest and al. (2011), Pink and dotted lines: our estimator with two differentiation techniques. CNAM seminar, february /35

20 Self-nested diagonals Non-parametric estimation Parametric estimation Confidence bands for δ 1 imply confidence bands for φ Figure : (Left) Confidence bands for δ 1 and δ 1 for chosen parameters α = β = 1.05, α + = β + = 0.9. (Right) Resulting theoretical confidence band for φ. Here C is a Gumbel copula of parameter θ = 2, the size of generated sample is n = CNAM seminar, february /35

21 Part II. Estimation of the distorted Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation

22 A class of parametric transformation Self-nested diagonals Non-parametric estimation Parametric estimation We take back from Bienvenüe and R. (2012): Definition (Conversion and distortion functions) Let f any bijective increasing function from R to R. It is said to be a conversion function. The distortion T f : [0, 1] [0, 1] is defined as 0 if u = 0, T f (u) = logit 1 (f (logit(u))) if 0 < u < 1, 1 if u = 1. Remark: Distortions 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: ( ) f (x) = H m,h,ρ1,ρ 2,η (x) = m h + (eρ 1 + e ρ 2 ) x m h 2 (e ρ 1 e ρ 2 ) x m h e η ρ 1 +ρ 2 2 with m, h, ρ 1, ρ 2 R, and one smoothing parameter η R. H 1 m,h,ρ 1,ρ 2,η (x) = H m, h, ρ 1, ρ 2,η(x). CNAM seminar, february /35

23 Parametric distortions estimation Self-nested diagonals Non-parametric estimation Parametric estimation Framework: An initial given copula C 0 is distorted using a distortion T, Initial given margins are distorted using distortions T 1,..., T d. Estimation: Get non-parametric estimators for T, Get non-parametric estimators for T 1,..., T d, Fit all distortions T, T 1,..., T d by piecewise-linear functions (e.g. in the logit scale), Approach piecewise linear functions by composition of hyperbolas, cf. Bienvenüe and R. (2012). CNAM seminar, february /35

24 Self-nested diagonals Non-parametric estimation Parametric estimation Geyser data: distorted 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 distorted density f (x 1, x 2 ) and Old Faithful geyser data (red points. CNAM seminar, february /35

25 Part III. Definitions Transformed Archimedean copulas Estimation given tail coefficients

26 multivariate TDC Definition Transformed Archimedean copulas Estimation with given tail coefficients Definition (Multivariate tail dependence coefficients: 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 coefficients is given by De Luca and Riviecio, 2012 (when limits exist): λ I h,īh L = lim λi h,ī h u 0 + L (u) with λ I h,īh L (u) = P [ ] U i u, i I h U i u, i Ī h, λ I h,īh U = lim λi h,ī h u 1 U (u) with λ I h,īh U (u) = P [ ] U i u, i I h U i u, i Ī h. If for all I h I, λ I h,ī h L = 0, (resp. λ I h,ī h U = 0) then we say U is lower tail independent (resp. upper tail independent). CNAM seminar, february /35

27 Definition Transformed Archimedean copulas Estimation with given tail coefficients 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. Coeffs. for Archimedean copulas) For Archimedean copulas the multivariate lower and upper tail dependence coefficients are respectively (cf. De Luca and Riviecio, 2012): 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 i=0 ( 1)i Cd i ψ 1 (i ψ(u)) d h i=0 ( 1)i Cd h i ψ 1 (i ψ(u)). Some references on tail dependence and Archimedean copulas: Juri and Wüthrich (2003), Charpentier and Segers (2009), Durante and al (2010). CNAM seminar, february /35

28 TDC for some usual copulas Definition Transformed Archimedean copulas Estimation with given tail coefficients Clatyton copulas Gumbel copulas u Figure : Shape of the 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), Gumbel copulas (right panel). u CNAM seminar, february /35

29 Regular Variation Definition Transformed Archimedean copulas Estimation with given tail coefficients At infinity: 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, f RV α (0) f I RV α ( ) s > 0, lim x 0 + f (s x) f (x) = sα. At one, f RV α (1) f M I RV α ( ) s > 0, lim x 0 + f (1 s x) f (1 x) = sα. CNAM seminar, february /35

30 Part III. Definitions Transformed Archimedean copulas Estimation given tail coefficients

31 Considered transformations Definition Transformed Archimedean copulas Estimation with given tail coefficients 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, (1) The transformation T f have 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 R.. CNAM seminar, february /35

32 Definition Transformed Archimedean copulas Estimation with given tail coefficients 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 distorted generator will depend on properties of f, φ 0 and G. CNAM seminar, february /35

33 Regular generators Definition Transformed Archimedean copulas Estimation with given tail coefficients phi(x) upper lower psi(t) lower upper x t Figure : Generators φ Gumbel(θ) = exp( x 1/θ ) (left) and its inverse ψ Gumbel(θ) = ( ln t) θ (right) for a Gumbel copula with parameters θ = 4 (dashed lines), θ = 3 (full lines) and θ = 2 (dotted lines). CNAM seminar, february /35

34 Lower tail assumptions Definition Transformed Archimedean copulas Estimation with given tail coefficients Assumption (Lower-tails: assumptions on f, φ 0, G) Assume that f, φ 0 and G are continuous and differentiable functions, strictly monotone with respective proper inverse functions denoted f 1, ψ 0 = φ 1 0 and G 1. Furthermore, i) The function f has a left asymptote f (x) = ax + 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, + ). CNAM seminar, february /35

35 Upper tail assumptions Definition Transformed Archimedean copulas Estimation with given tail coefficients Assumption (Upper-tails: assumptions on f, φ 0, G) Assume that f, φ 0 and G are continuous and differentiable functions, strictly monotone with respective proper inverse functions denoted f 1, ψ 0 = φ 1 0 and G 1. Furthermore, i) The function f has a right asymptote f (x) = αx + 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, + ). CNAM seminar, february /35

36 TDC of distorted generators Definition Transformed Archimedean copulas Estimation with given tail coefficients Theorem (Multivariate lower TDC of transformed Archimedean copula) If f has a left asymptote ax + b, a (0, + ), b (, + ), ψ 0 RV r0 (0), r 0 [0, + ], and the hazard rate m G = G /G RV g 1 ( ), for g (0, + ). Then, transformed lower TDC is λ (h,d h) L = see in two slides, if r 0 = 0, d ag r 1 0 (d h) ag r 1 0, if r 0 (0, + ), 1, if r 0 = +. (2) CNAM seminar, february /35

37 TDC of distorted generators Definition Transformed Archimedean copulas Estimation with given tail coefficients Theorem (Multivariate Upper TDC of transformed Archimedean copula) If f has a right asymptote α x + β, α (0, + ), β (, + ), ψ 0 RV ρ0 (1), ρ 0 [1, + ], and the hazard rate µ G = G /Ḡ RV γ 1 ( ), γ (0, + ). Then, when ρ = ρ 0 α γ 1, transformed upper TDC is λ (h,d h) U = see next slide, if ρ 0 = 1, d i =1 C i d ( 1)i i αγ ρ 1 0 d h i =1 C i d h ( 1)i i αγ ρ 1 0, if ρ 0 (1, + ), 1, if ρ 0 = +. (3) CNAM seminar, february /35

38 Definition Transformed Archimedean copulas Estimation with given tail coefficients Asymptotic lower tail independence When ψ 0 RV r0 (0), with r 0 = 0, λ (h,d h) L = lim λ(h,d h) u 0 + L (u) = 0 if µ φ0 = φ 0 /φ0 RV k 0 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 differentiable generator. λ (h,d h) U = lim u 1 λ(h,d h) U (u) = 0. if ( D) d φ(0) is finite 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). CNAM seminar, february /35

39 Part III. Definitions Transformed Archimedean copulas Estimation given tail coefficients

40 Fit with given TDC Definition Transformed Archimedean copulas Estimation with given tail coefficients In the bivariate case, when d = 2, λ (1,1) L = 2 ag r 1 0 (1,1) and λ U = 2 2 αγ ρ 1 0 Take a distortion T f (x) = G f G 1 (x) with f = H m, h, p1, p 2, η, H m, h, p1, p 2, 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, One can deduce two parameters ( ) p 1 = 1 (1,1) g ln ln λ L r 0 ln 2 and p 2 = 1 γ ln ( ln(2 ρ 0 ln 2 (1,1) λ U ) ). (4) In the multivariate case, when d > 2, same principle, but expression of p 2 more difficult to write. CNAM seminar, february /35

41 distorted copulas with chosen TDC Definition Transformed Archimedean copulas Estimation with given tail coefficients Models A, B, C, D u Figure : Concentration function λ LU (u) = 1 {u 1/2} λ L (u) + 1 {u>1/2} λ U (u) for some distorted copulas. Chosen values: λ L = 1/4 and λ U = 3/4. CNAM seminar, february /35

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43 On the non-parametric estimation of the generator: Comparable performance with other estimators, Without solving non-linear systems of equations, Theoretical confidence bands. Class of estimators that depend on the initial copula. On the parametric estimation of the generator: Analytical expressions for the level curves, Tunable number of parameters, can be chosen Among further perspectives: Properties of the estimator (convexity, etc.) Estimators using other informations Use with nested copulas CNAM seminar, february /35

44 Thank you for your attention. CNAM seminar, february /35

45 papers that were 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. (2014). Estimation of multivariate critical layers: Applications to rainfall data. Preprint available on HAL (open access french archive website). more details on part: Di Bernardino, E. and Rullière, D. (2014). On tail dependence coefficients of transformed multivariate Archimedean copulas. Preprint available on HAL (open access french archive website). CNAM seminar, february /35