Non parametric estimation of Archimedean copulas and tail dependence Elena Di Bernardino a and Didier Rullière b Paris, february 19, 2015. a CNAM, Paris, Département IMATH, b ISFA, Université Lyon 1, Laboratoire SAF. CNAM seminar, february 2015 1/35
Part I. Transformations of Archimedean copulas
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 2015 2/35
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 2015 3/35
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 2015 4/35
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 2015 5/35
Part II. Estimation of the distorted Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation
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 2015 6/35
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 2015 7/35
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 2015 8/35
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 2015 9/35
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 2015 10/35
Part II. Estimation of the distorted Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation
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 2015 11/35
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 2015 12/35
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 2015 13/35
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 2015 14/35
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 2015 15/35
Comparison with other estimators Self-nested diagonals Non-parametric estimation Parametric estimation lambda 0.12 0.10 0.08 0.06 0.04 0.02 0.00 tau = 0.75, n=1000, d=2 0.0 0.2 0.4 0.6 0.8 1.0 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 2015 16/35
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 = 2000. CNAM seminar, february 2015 17/35
Part II. Estimation of the distorted Archimedean copula Self-nested diagonals Non-parametric estimation Parametric estimation
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 2 2 + 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 2015 18/35
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 2015 19/35
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 2015 20/35
Part III. Definitions Transformed Archimedean copulas Estimation given tail coefficients
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 2015 21/35
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 2015 22/35
TDC for some usual copulas Definition Transformed Archimedean copulas Estimation with given tail coefficients Clatyton copulas Gumbel copulas 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 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 2015 23/35
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 2015 24/35
Part III. Definitions Transformed Archimedean copulas Estimation given tail coefficients
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 2015 25/35
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 2015 26/35
Regular generators Definition Transformed Archimedean copulas Estimation with given tail coefficients phi(x) 0.0 0.2 0.4 0.6 0.8 1.0 upper lower psi(t) 0 1 2 3 4 5 6 7 lower upper 0 20 40 60 80 100 x 0.2 0.4 0.6 0.8 1.0 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 2015 27/35
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 2015 28/35
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 2015 29/35
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 2015 30/35
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 2015 31/35
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 2015 32/35
Part III. Definitions Transformed Archimedean copulas Estimation given tail coefficients
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 2015 33/35
distorted copulas with chosen TDC Definition Transformed Archimedean copulas Estimation with given tail coefficients Models A, B, C, D 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 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 2015 34/35
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 2015 35/35
Thank you for your attention. CNAM seminar, february 2015 36/35
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): 1-36. 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):190-205. 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 2015 37/35