Copulas: An Introduction Part II: Models Johan Segers Universit - PowerPoint PPT Presentation

Copulas: An Introduction Part II: Models Johan Segers Université catholique de Louvain (BE) Institut de statistique, biostatistique et sciences actuarielles Columbia University, New York City 9–11 Oct 2013 Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 1 / 65

Copulas: An Introduction Part II: Models Archimedean copulas Extreme-value copulas Elliptical copulas Vines Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 2 / 65

Copulas: An Introduction Part II: Models Archimedean copulas Extreme-value copulas Elliptical copulas Vines Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 3 / 65

The (in)famous Archimedean copulas ◮ By far the most popular (theory & practice) class of copulas ◮ Plenty of parametric models ◮ Gumbel, Clayton, Frank, Joe, Ali–Mikhail–Haq, . . . ◮ Building block for more complicated constructions: ◮ Nested/Hierarchical Archimedean copulas ◮ Vine copulas ◮ Archimax copulas ◮ . . . ◮ Mindless application of (Archimedean) copulas has drawn many criticisms on the copula ‘hype’ Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 4 / 65

Laplace transform of a positive random variable Recall the Laplace transform of a random variable Z > 0: � ∞ e − sz d F Z ( z ) , ψ ( s ) = E [ exp ( − sZ )] = s ∈ [ 0 , ∞ ] 0 A distribution on ( 0 , ∞ ) is identified by its Laplace transform. Ex. Show the following properties: ◮ 0 ≤ ψ ( s ) ≤ 1 ◮ ψ ( 0 ) = 1 and ψ ( ∞ ) = 0 . ◮ ( − 1 ) k d k ψ ( s ) / d s k ≥ 0 for all integer k ≥ 1 . ◮ In particular, ψ is nonincreasing ( k = 1 ) and convex ( k = 2 ). Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 5 / 65

Survival functions in proportional hazards model: The Laplace transform of the frailty appears Independent unit exponential random variables Y 1 , . . . , Y d . Survival times X 1 , . . . , X d are affected by a common ‘frailty’ Z > 0: X j = Y j / Z Marginal and joint survival functions: Pr [ X j > x j ] = E [ e − x j Z ] = ψ ( x j ) Pr [ X 1 > x 1 , . . . , X d > x d ] = E [ e − ( x 1 + ··· + x d ) Z ] = ψ ( x 1 + · · · + x d ) Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 6 / 65

In proportional hazards models, survival copulas are Archimedean The survival copula of X is Archimedean with generator ψ : � � ¯ ψ − 1 ( u 1 ) + · · · + ψ − 1 ( u d ) C ( u 1 , . . . , u d ) = ψ Ex. Show the above formula. Ex. Show that replacing Z by β Z for a constant β > 0 changes ψ but does not change the copula. Ex. Pick your favourite (discrete/continuous) distribution on ( 0 , ∞ ) , compute or look up its Laplace transform, and compute the associated Archimedean copula. If it doesn’t exist yet, name it after yourself and publish a paper about it. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 7 / 65

A Gamma frailty induces the Clayton copula If Z ∼ Gamma ( 1 /θ, 1 ) , with 0 < θ < ∞ , then � ∞ e − sz z 1 /θ − 1 e − z d z = ( 1 + s ) − 1 /θ ψ ( s ) = Γ( 1 /θ ) 0 and the resulting survival copula is Clayton: C ( u ) = ( u − θ ¯ + · · · + u − θ − d + 1 ) − 1 /θ 1 d Ex. Check the above formulas. Ex. How to use the frailty representation to sample from a Clayton copula? Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 8 / 65

Generator of the Clayton copula Generator Inverse generator w = ψ ( s ) s = ψ − 1 ( w ) 2.0 2.0 1.5 1.5 w 1.0 1.0 s 0.5 0.5 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 s w s = ψ − 1 ( w ) = w − θ − 1 w = ψ ( s ) = ( 1 + s ) − 1 /θ Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 9 / 65

Formal definition of an Archimedean copula A copula C is Archimedean if there exists ψ : [ 0 , ∞ ] → [ 0 , 1 ] such that � � ψ − 1 ( u 1 ) + · · · + ψ − 1 ( u d ) C ( u ) = ψ For C to be a copula, it is sufficient and necessary that ψ satisfies ◮ ψ ( 0 ) = 1 and ψ ( ∞ ) = 0 ◮ ψ is d -monotone, i.e. ◮ ( − 1 ) k d k ψ ( s ) / d s k ≥ 0 for k ∈ { 0 , . . . , d − 2 } ◮ ( − 1 ) d − 2 d d − 2 ψ ( s ) / d s d − 2 is decreasing and convex Equivalently, there should exists a random variable Z > 0 such that �� � d − 1 � sZ ψ ( s ) = E 1 − d − 1 + i.e. ψ is the Williamson d -transform of the rv ( d − 1 ) / Z . Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 10 / 65

Standard examples Ex. The independence copula Π( u ) = u 1 · · · u d is Archimedean. ◮ What is its generator ψ ? ◮ What is the frailty variable Z ? Ex. The Fréchet–Hoeffding lower bound W ( u , v ) = max ( u + v − 1 , 0 ) is Archimedean too. What is its generator ψ ? [This ψ is not a Laplace transform; it is 2 -monotone but not d -monotone for d ≥ 3 .] Ex. One can show that the Fréchet–Hoeffding upper bound M ( u ) = min ( u 1 , . . . , u d ) is not Archimedean. Still, show that the Clayton copula with θ → ∞ converges to M . Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 11 / 65

Common generator functions w = ψ ( s ) w = ψ ( s ) 2.0 2.0 1.5 1.5 1.0 1.0 w w 0.5 0.5 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 s s Π( u ) = u 1 · · · u d W ( u , v ) = max ( u + v − 1 , 0 ) ψ ( s ) = e − s ψ ( s ) = max ( 1 − s , 0 ) Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 12 / 65

Bivariate Archimedean copulas as binary operators A bivariate Archimedean copula induces a binary operator [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , 1 ] : ( u , v ) �→ C ( u , v ) which is commutative and associative: C ( u , v ) = C ( v , u ) , C ( u , C ( v , w )) = C ( C ( u , v ) , w ) endowing [ 0 , 1 ] with a semi-group structure. Link with the theory of associative functions (A BEL , H ILBERT ). Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 13 / 65

Derived quantities Conditional cdf: C j ( u ) = ψ ′ � � ψ − 1 ( u 1 ) + · · · + ψ − 1 ( u d ) ˙ ψ ′ � � ψ − 1 ( u j ) Pdf, provided ψ is d times continuously differentiable c ( u ) = ψ ( d ) � � ψ − 1 ( u 1 ) + · · · + ψ − 1 ( u d ) � d j = 1 ψ ′ � � ψ − 1 ( u j ) Ex. Show these formulas. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 14 / 65

Yet another probability integral transform: Kendall distribution functions Bivariate cdf H , continuous margins F and G , copula C . The Kendall distribution of a random pair ( X , Y ) ∼ H is the cdf of the rv W = H ( X , Y ) = C ( F ( X ) , G ( Y )) = C ( U , V ) It only depends on H through C : � K C ( w ) = Pr ( W ≤ w ) = [ 0 , 1 ] 2 1 { C ( u , v ) ≤ w } d C ( u , v ) , w ∈ [ 0 , 1 ] It is linked to Kendall’s tau via � 1 � [ 0 , 1 ] 2 C ( u , v ) d C ( u , v ) = 1 + τ E [ W ] = w d K C ( w ) = 4 0 Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 15 / 65

Kendall distribution functions: The C -probability below a C -level curve contour plot of C(u, v) = uv 1.0 0.9 0.8 0.8 0.7 0 0.6 . 6 K ( w ) = 0.5 v � 0.4 0 . 4 [ 0 , 1 ] 2 1 { C ( u , v ) ≤ w } d C ( u , v ) 0.3 0.2 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 u Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 16 / 65

Bivariate Archimedean copulas are identified by their Kendall distribution function The Kendall distribution function of a bivariate Archimedean copula with inverse generator φ = ψ − 1 : ( 0 , 1 ] → [ 0 , ∞ ) is K ( w ) = w − λ ( w ) , λ ( w ) = φ ( w ) 1 φ ′ ( w ) = d log φ ( w ) / d w ≤ 0 Up to a multiplicative constant, φ and thus ψ can be reconstructed from λ . Ex. Show the following properties: ◮ K Π ( w ) = w − w log ( w ) (independence) ◮ K W ( w ) = 1 (Fréchet–Hoeffding lower bound) ◮ K M ( w ) = w (Fréchet–Hoeffding upper bound) ◮ w ≤ K ( w ) ≤ 1 Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 17 / 65

Kendall distribution functions: Stochastically smaller than the uniform one Kendall distribution function 1.0 0.8 0.6 K(w) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 w Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 18 / 65

The tail behaviour of a bivariate Archimedean copula can be read off from the inverse generator function Coefficient of lower tail dependence: C ( w , w ) = 2 − 1 /θ 0 , λ L ( C ) = lim w w ↓ 0 w φ ′ ( w ) where θ 0 = − lim ∈ [ 0 , ∞ ] φ ( w ) w ↓ 0 Coefficient of upper tail dependence: λ U ( C ) = λ L (¯ C ) = 2 − 2 1 /θ 1 , w φ ′ ( 1 − w ) where θ 1 = − lim ∈ [ 1 , ∞ ] φ ( 1 − w ) w ↓ 0 ⇒ Construction of models with different upper and lower tails Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 19 / 65

Archimedean copulas enjoy many symmetries Let ( U 1 , . . . , U d ) ∼ C and C is Archimedean with generator ψ . ◮ Permuation symmetry: For any permuation σ of { 1 , . . . , d } , ( U σ ( 1 ) , . . . , U σ ( d ) ) d = ( U 1 , . . . , U d ) ◮ Closure of margins: For any subset 1 ≤ j 1 < · · · < j k ≤ d , ( U j 1 , . . . , U j k ) ∼ k -variate Archimedean, same generator ψ Symmetry is a blessing (simplicity) and a curse (lack of flexibility). The only radially symmetric Archimedean copula ( C = ¯ C ) is the Frank copula. Johan Segers (UCL) Copulas. II - Models Columbia University, Oct 2013 20 / 65