Sampling Archimedean copulas
Sampling Archimedean copulas
Marius Hofert [email protected]
Ulm University 2007-12-08
Sampling Archimedean copulas
Outline 1 Nonnested Archimedean copulas 1.1 Marshall and Olkin’s algorithm 2 Nested Archimedean copulas 2.1 McNeil’s algorithm 2.2 A general nesting result 2.3 Special nesting results 2.4 Simulation results
2
Sampling Archimedean copulas
1 Nonnested Archimedean copulas k
d −1 (t) ≥ 0) • Kimberling: ϕ−1 c.m. (i.e. (−1)k dt kϕ
⇔ ϕ−1 [ϕ(u1 ) + · · · + ϕ(ud )] a copula. • Sampling ϕ−1 [ϕ(u1 ) + · · · + ϕ(ud )]: – Via Conditional distribution method (density known). – Via Laplace Stieltjes transforms. • Bernstein: ϕ−1 c.m. and ϕ−1 (0) = 1 ⇔ ϕ−1 = LS(FV (x)), supp(FV ) ⊂ [0, ∞).
3
Sampling Archimedean copulas
1.1 Marshall and Olkin’s algorithm Algorithm (Marshall, Olkin) (1) Sample V ∼ FV . (2) Sample i.i.d. realizations Xi ∼ U [0, 1], i ∈ {1, . . . , d}. (3) Return (U1 , . . . , Ud ), where Ui = ϕ−1 (− log(Xi )/V ). ⇒ Fast and easy (if FV is easy to sample). Examples for FV : Pn • AMH: k=1 (1 − ϑ)ϑk−1 , n ∈ N, ϑ ∈ [0, 1) ⇒ Geometric. Pn (1−e−ϑ )k , n ∈ N, ϑ ∈ (0, ∞) ⇒ Logarithmic. • Frank: k=1 kϑ Pn 1/ϑ k+1 • Joe: k=1 (−1) k , n ∈ N, ϑ ∈ [1, ∞). 4
Sampling Archimedean copulas
• Clayton: Γ(1/ϑ, 1), ϑ ∈ (0, ∞) ⇒ Gamma. π • Gumbel: S(1/ϑ, 1, (cos( 2ϑ ))ϑ , 0; 1), ϑ ∈ [1, ∞) ⇒ Stable.
Theorem P∞ −1 Let ϕ = LS(FV ) and F (x) = k=0 yk 1[xk ,∞) (x) with P∞ 0 < x0 < x1 < . . . and yk ≥ 0, k ∈ N0 , with k=0 yk = 1. Then FV ≡ F ⇔ ϕ−1 (t) =
∞ X k=0
5
yk e−xk t .
Sampling Archimedean copulas
2 Nested Archimedean copulas • Now: C(u) = ϕ−1 [ϕ(u1 ) + ϕ(ϕ−1 1 [ϕ1 (u2 ) + ϕ1 (u3 )])] – Fully vs. partially nested Archimedean copulas. – Nonexchangeable (max. d − 1 different pairwise dependencies). – Allows for modeling different industry sectors, regions, etc. ′ • McNeil: Nesting condition (ϕ ◦ ϕ−1 1 ) c.m. ⇒ C(u) is a copula.
•
′ (ϕ ◦ ϕ−1 ) 1
ϕ−1 0,1 (t; v)
−1 −vϕ◦ϕ 1 (t) e
c.m. ⇒ is a generator inverse for = every v ∈ (0, ∞) with corresponding inner FV = LS −1 (ϕ−1 0,1 ).
6
Sampling Archimedean copulas
2.1 McNeil’s algorithm Algorithm (McNeil) (1) Sample X1 ∼ U [0, 1]. (2) Sample V ∼ FV = LS −1 (ϕ−1 ), (X2 , X3 ) ∼ C(u2 , u3 ; ϕ−1 0,1 (·; V )). (3) Return (U1 , U2 , U3 ), where Ui = ϕ−1 (− log(Xi )/V ). Proof Solving w.r.t. the X’s and conditioning under V leads to Z ∞ −vϕ(u2 ) −vϕ(u3 ) (e ) + ϕ (e )] dFV (v) P(U ≤ u) = e−vϕ(u1 ) ϕ−1 [ϕ 0,1 0,1 0,1 Z0 ∞ −vϕ(u1 ) −vϕ(ϕ−1 1 [ϕ1 (u2 )+ϕ1 (u3 )]) dF (v) = e e V 0 −1
=ϕ
[ϕ(u1 ) + ϕ(ϕ−1 1 [ϕ1 (u2 ) + ϕ1 (u3 )])]. 7
Sampling Archimedean copulas
Examples: • Nested Gumbel copula: ϑ/ϑ
1 −1 ϑ1 /ϑ belongs to −vt – ϕ−1 ⇒ ϕ v) with c = v (t; v) = e (t/c; 0,1 0,1 the Gumbel family again.
– Sample C(u2 , u3 ; ϕ−1 0,1 (·/c; V )) (involves Stable distribution). • Nested Clayton copula: –
ϕ−1 0,1 (t; v)
−v((1+t)ϑ/ϑ1 −1)
=e large enough ϑ).
⇒ Rejection algorithm (only for
Questions: • Which (classes of) generators can be used to build a nested structure? 8
Sampling Archimedean copulas
• Corresponding sampling strategies? • Fast in large dimensions? 2.2 A general nesting result • Power families (d = 2): ˜ = ϕ(tα ) for any α ∈ (0, 1]. – Inner power families: ϕ(t) – Outer power families: ϕ(t) ˜ = ϕ(t)β for any β ∈ [1, ∞). ⇒ Can they be nested and easily sampled? • For inner power families, the sufficient nesting condition is usually not fulfilled (e.g. take AMH based on ϑ = 1/2).
9
Sampling Archimedean copulas
Theorem (a) For an Archimedean generator ϕ with c.m. inverse, ϕ(t) ˜ = (c + ϕ(t))ϑ − cϑ has a c.m. inverse for any ϑ ∈ [1, ∞) and c ∈ [0, ∞). (b) The sufficient nesting condition holds for any ϑ ≤ ϑ1 and the inner −1 ϑ1 +t)ϑ/ϑ1 −cϑ ) −v ϕ◦ ˜ ϕ ˜ 1 (t) = e−v((c . generator inverse is given by e • Families # 1, 4, 12, 13 and 19 of Nelson (1998) fall under this setup. • The inner FV is an exponentially tilted Stable distribution (c = 1 ⇒ Clayton). • For outer power families (c = 0) we obtain: They extend to d > 2 and they can be nested. 10
Sampling Archimedean copulas
• The inner FV is a Stable distribution, independent of the family on which the outer power family is built! • For the outer FV , the following algorithm generates a V˜ ∼ FV˜ = LS −1 (ϕ˜−1 ), where ϕ(t) ˜ = ϕ(t)ϑ , ϑ ∈ [1, ∞). Algorithm (1) Sample V ∼ FV = LS −1 (ϕ−1 ). π (2) Sample S ∼ S(1/ϑ, 1, (cos( 2ϑ ))ϑ , 0; 1).
(3) Return SV ϑ . Proof R ∞ R ∞ −tvϑ s fS (s) ds dFV (v) (LS(FSV ϑ ))(t) = 0 0 e R ∞ −(tvϑ )1/ϑ dFV (v) = (LS(FV ))(t1/ϑ ) = ϕ˜−1 (t). = 0 e 11
Sampling Archimedean copulas
⇒ Knowing FV = LS −1 (ϕ−1 ), we can build and easily sample an outer power nested Archimedean copula based on ϕ. Theorem Kendall’s tau τϕ˜ , belonging to the copula C˜ generated by ϕ(t) ˜ = ϕ(t)ϑ , ϑ ∈ [1, ∞), is given by 1 τϕ˜ = 1 − (1 − τϕ ). ϑ Example: A nested outer power Clayton copula • Outer power family build on Clayton’s generator ϕ(t) = t−ϑc − 1. ⇒ τϕ˜ = 1 − ϑ1 ϑc2+2 , λL,ϕ˜ = 2−1/(ϑc ϑ) and λU,ϕ˜ = 2 − 21/ϑ
12
Sampling Archimedean copulas
• Scatterplot matrix for a fully nested outer power Clayton copula (ϑc = 1, ϑ = 1.1, ϑ1 = 1.5):
Component 3
Component 2
Component 1
13
Sampling Archimedean copulas
2.3 Special nesting results Discrete distributions • For AMH, Frank and Joe: outer FV discrete. ⇒ What about the inner FV ’s? Theorem If ϕ and ϕ1 denote Joe’s generators with ϑ ≤ ϑ1 , then ϕ−1 0,1 (t; v), v ∈ N, P∞ has Laplace Stieltjes inverse FV (x) = k=1 yk 1[xk ,∞) (x) with v X jϑ/ϑ1 j+k v xk = k and yk = , k ∈ N. (−1) j k j=0
• Precalculation numerically complicated: Runtime (dependence on v sample) and errors (FV (k) = 0, k < v and FV (v) = (ϑ/ϑ1 )v ). 14
Sampling Archimedean copulas
⇒ Not adequate for sampling purposes! Idea Use the generating function for the inner FV : g(x) = (1 − (1 − x)ϑ/ϑ1 )v .
Lemma (Devroye) Let g, g1 , g2 be g.f.’s such that g(t) = g1 (g2 (t)) and let N and X have g.f. g1 and g2 , respectively, then Y =
N X
Xi has generating function g,
i=1
where the Xi ’s are i.i.d. copies of X. 15
Sampling Archimedean copulas
Algorithm (1) Precalculate the outer FV once (yk =
1/ϑ k−1 (−1) , k
k ∈ N).
(2) Precalculate F corresponding to the g.f. 1 − (1 − x)ϑ/ϑ1 once ϑ/ϑ1 (yk = k (−1)k−1 , k ∈ N).
(3) Sample V ∼ FV .
(4) Sample i.i.d. Xi ∼ F , i ∈ {1, . . . , V } and build of the inner FV ).
PV
i=1 Xi
(a sample
(5) Proceed as before. • For Frank’s nested family: Similar. • For AMH’s nested family: Inner FV also a geometric distribution (no precalculation). 16
Sampling Archimedean copulas
• Scatterplot matrix for a fully nested Joe copula (ϑ = 1.2, ϑ1 = 1.6):
Component 3
Component 2
Component 1
17
Sampling Archimedean copulas
Mixed families • Is it possible to mix different families? • Examples from Nelson (1998): The sufficient nesting condition holds for 7 family combinations including (AMH,Clayton) for ϑ ∈ [0, 1), ϑ1 ∈ [1, ∞). A nested (AMH,Clayton) copula • Parameters ϑ = 0.8 and ϑ1 = 2. • Outer FV : Known geometric distribution. 1/ϑ1 + ϑ)−v ⇒ Inner F not • Inner FV : ϕ−1 V 0,1 (t; v) = ((1 − ϑ)(1 + t) explicitly known.
18
Sampling Archimedean copulas
Idea Apply numerical inversion of Laplace transforms and numerical rootfinding to obtain a realization from FV (x) = (L−1 (ϕ−1 (t)/t))(x), x ∈ [0, ∞). • Methods: – Fixed Talbot algorithm (contour deformation in the Bromwich integral). – Gaver Stehfest and Gaver Wynn rho algorithm (discrete analog of the Post-Widder formula). – Laguerre series algorithm (approximation via Laguerres series). • Comparison in the known nonnested Clayton case (ϑ = 0.8):
19
Sampling Archimedean copulas
Parameter calibration according to the precision requirement F (x) − F˜ (x) V V max < 0.0001 for P = {0.001, 0.002, . . . , 10}. x∈P FV (x)
for an approximation F˜V to FV . 2.4 Simulation results
• χ2 -test based on hypercubes of a partition into 5 parts for each dimension. • Maximal (MXDEV) and mean (MDEV) deviations of the probabilities of falling in the hypercubes. • Matrix of pairwise sample versions of Kendall’s tau. • Visual check of plots of outer and inner FV ’s. 20
Sampling Archimedean copulas
For the nested (AMH,Clayton) copula Based on 500.000 observations, ϑ = 0.8 and ϑ1 = 2, using the Fixed Talbot algorithm: • p-value = 1.000000000000 (χ2 -test). • MXDEV = 0.000555 and MDEV = 0.000084. • τˆϑ ∈ {0.2346, 0.2343} (τϑ = 0.2337) and τˆϑ1 = 0.5009 (τϑ1 = 0.5).
21
Sampling Archimedean copulas
1.0
1.0
• Plots for the outer (left) and inner (right) distribution functions FV :
(1) (2)
0.8 0.6
(4)
0.4 0.2
Value 0.4 0.2
Value
0.6
0.8
(3)
theta=0.8, theta2=2.0, v=0.1 theta=0.8, theta2=2.0, v=0.5 theta=0.8, theta2=2.0, v=1 theta=0.8, theta2=2.0, v=2
0.0
0.0
(1) (2) (3) (4)
0
5
10
15
20
0.0
Variable
0.2
0.4
0.6
Variable
22
0.8
1.0
Sampling Archimedean copulas
• Scatterplot matrix for this fully nested (AMH,Clayton) copula (ϑ = 0.8, ϑ1 = 2):
Component 3
Component 2
Component 1
23
Sampling Archimedean copulas
Runtimes For 500.000 observations of ϕ−1 [ϕ(u1 ) + ϕ(ϕ−1 1 [ϕ1 (u2 ) + ϕ1 (u3 )])], we obtain: • (AMH,Clayton): 37.67s. • Gumbel: 2.22s. Outer power Clayton: 2.71s. • Joe: 1.50s. Frank: 1.60s. AMH: 0.59s.
24
Marius Hofert [email protected]
Ulm University 2007-12-08
Sampling Archimedean copulas
Outline 1 Nonnested Archimedean copulas 1.1 Marshall and Olkin’s algorithm 2 Nested Archimedean copulas 2.1 McNeil’s algorithm 2.2 A general nesting result 2.3 Special nesting results 2.4 Simulation results
2
Sampling Archimedean copulas
1 Nonnested Archimedean copulas k
d −1 (t) ≥ 0) • Kimberling: ϕ−1 c.m. (i.e. (−1)k dt kϕ
⇔ ϕ−1 [ϕ(u1 ) + · · · + ϕ(ud )] a copula. • Sampling ϕ−1 [ϕ(u1 ) + · · · + ϕ(ud )]: – Via Conditional distribution method (density known). – Via Laplace Stieltjes transforms. • Bernstein: ϕ−1 c.m. and ϕ−1 (0) = 1 ⇔ ϕ−1 = LS(FV (x)), supp(FV ) ⊂ [0, ∞).
3
Sampling Archimedean copulas
1.1 Marshall and Olkin’s algorithm Algorithm (Marshall, Olkin) (1) Sample V ∼ FV . (2) Sample i.i.d. realizations Xi ∼ U [0, 1], i ∈ {1, . . . , d}. (3) Return (U1 , . . . , Ud ), where Ui = ϕ−1 (− log(Xi )/V ). ⇒ Fast and easy (if FV is easy to sample). Examples for FV : Pn • AMH: k=1 (1 − ϑ)ϑk−1 , n ∈ N, ϑ ∈ [0, 1) ⇒ Geometric. Pn (1−e−ϑ )k , n ∈ N, ϑ ∈ (0, ∞) ⇒ Logarithmic. • Frank: k=1 kϑ Pn 1/ϑ k+1 • Joe: k=1 (−1) k , n ∈ N, ϑ ∈ [1, ∞). 4
Sampling Archimedean copulas
• Clayton: Γ(1/ϑ, 1), ϑ ∈ (0, ∞) ⇒ Gamma. π • Gumbel: S(1/ϑ, 1, (cos( 2ϑ ))ϑ , 0; 1), ϑ ∈ [1, ∞) ⇒ Stable.
Theorem P∞ −1 Let ϕ = LS(FV ) and F (x) = k=0 yk 1[xk ,∞) (x) with P∞ 0 < x0 < x1 < . . . and yk ≥ 0, k ∈ N0 , with k=0 yk = 1. Then FV ≡ F ⇔ ϕ−1 (t) =
∞ X k=0
5
yk e−xk t .
Sampling Archimedean copulas
2 Nested Archimedean copulas • Now: C(u) = ϕ−1 [ϕ(u1 ) + ϕ(ϕ−1 1 [ϕ1 (u2 ) + ϕ1 (u3 )])] – Fully vs. partially nested Archimedean copulas. – Nonexchangeable (max. d − 1 different pairwise dependencies). – Allows for modeling different industry sectors, regions, etc. ′ • McNeil: Nesting condition (ϕ ◦ ϕ−1 1 ) c.m. ⇒ C(u) is a copula.
•
′ (ϕ ◦ ϕ−1 ) 1
ϕ−1 0,1 (t; v)
−1 −vϕ◦ϕ 1 (t) e
c.m. ⇒ is a generator inverse for = every v ∈ (0, ∞) with corresponding inner FV = LS −1 (ϕ−1 0,1 ).
6
Sampling Archimedean copulas
2.1 McNeil’s algorithm Algorithm (McNeil) (1) Sample X1 ∼ U [0, 1]. (2) Sample V ∼ FV = LS −1 (ϕ−1 ), (X2 , X3 ) ∼ C(u2 , u3 ; ϕ−1 0,1 (·; V )). (3) Return (U1 , U2 , U3 ), where Ui = ϕ−1 (− log(Xi )/V ). Proof Solving w.r.t. the X’s and conditioning under V leads to Z ∞ −vϕ(u2 ) −vϕ(u3 ) (e ) + ϕ (e )] dFV (v) P(U ≤ u) = e−vϕ(u1 ) ϕ−1 [ϕ 0,1 0,1 0,1 Z0 ∞ −vϕ(u1 ) −vϕ(ϕ−1 1 [ϕ1 (u2 )+ϕ1 (u3 )]) dF (v) = e e V 0 −1
=ϕ
[ϕ(u1 ) + ϕ(ϕ−1 1 [ϕ1 (u2 ) + ϕ1 (u3 )])]. 7
Sampling Archimedean copulas
Examples: • Nested Gumbel copula: ϑ/ϑ
1 −1 ϑ1 /ϑ belongs to −vt – ϕ−1 ⇒ ϕ v) with c = v (t; v) = e (t/c; 0,1 0,1 the Gumbel family again.
– Sample C(u2 , u3 ; ϕ−1 0,1 (·/c; V )) (involves Stable distribution). • Nested Clayton copula: –
ϕ−1 0,1 (t; v)
−v((1+t)ϑ/ϑ1 −1)
=e large enough ϑ).
⇒ Rejection algorithm (only for
Questions: • Which (classes of) generators can be used to build a nested structure? 8
Sampling Archimedean copulas
• Corresponding sampling strategies? • Fast in large dimensions? 2.2 A general nesting result • Power families (d = 2): ˜ = ϕ(tα ) for any α ∈ (0, 1]. – Inner power families: ϕ(t) – Outer power families: ϕ(t) ˜ = ϕ(t)β for any β ∈ [1, ∞). ⇒ Can they be nested and easily sampled? • For inner power families, the sufficient nesting condition is usually not fulfilled (e.g. take AMH based on ϑ = 1/2).
9
Sampling Archimedean copulas
Theorem (a) For an Archimedean generator ϕ with c.m. inverse, ϕ(t) ˜ = (c + ϕ(t))ϑ − cϑ has a c.m. inverse for any ϑ ∈ [1, ∞) and c ∈ [0, ∞). (b) The sufficient nesting condition holds for any ϑ ≤ ϑ1 and the inner −1 ϑ1 +t)ϑ/ϑ1 −cϑ ) −v ϕ◦ ˜ ϕ ˜ 1 (t) = e−v((c . generator inverse is given by e • Families # 1, 4, 12, 13 and 19 of Nelson (1998) fall under this setup. • The inner FV is an exponentially tilted Stable distribution (c = 1 ⇒ Clayton). • For outer power families (c = 0) we obtain: They extend to d > 2 and they can be nested. 10
Sampling Archimedean copulas
• The inner FV is a Stable distribution, independent of the family on which the outer power family is built! • For the outer FV , the following algorithm generates a V˜ ∼ FV˜ = LS −1 (ϕ˜−1 ), where ϕ(t) ˜ = ϕ(t)ϑ , ϑ ∈ [1, ∞). Algorithm (1) Sample V ∼ FV = LS −1 (ϕ−1 ). π (2) Sample S ∼ S(1/ϑ, 1, (cos( 2ϑ ))ϑ , 0; 1).
(3) Return SV ϑ . Proof R ∞ R ∞ −tvϑ s fS (s) ds dFV (v) (LS(FSV ϑ ))(t) = 0 0 e R ∞ −(tvϑ )1/ϑ dFV (v) = (LS(FV ))(t1/ϑ ) = ϕ˜−1 (t). = 0 e 11
Sampling Archimedean copulas
⇒ Knowing FV = LS −1 (ϕ−1 ), we can build and easily sample an outer power nested Archimedean copula based on ϕ. Theorem Kendall’s tau τϕ˜ , belonging to the copula C˜ generated by ϕ(t) ˜ = ϕ(t)ϑ , ϑ ∈ [1, ∞), is given by 1 τϕ˜ = 1 − (1 − τϕ ). ϑ Example: A nested outer power Clayton copula • Outer power family build on Clayton’s generator ϕ(t) = t−ϑc − 1. ⇒ τϕ˜ = 1 − ϑ1 ϑc2+2 , λL,ϕ˜ = 2−1/(ϑc ϑ) and λU,ϕ˜ = 2 − 21/ϑ
12
Sampling Archimedean copulas
• Scatterplot matrix for a fully nested outer power Clayton copula (ϑc = 1, ϑ = 1.1, ϑ1 = 1.5):
Component 3
Component 2
Component 1
13
Sampling Archimedean copulas
2.3 Special nesting results Discrete distributions • For AMH, Frank and Joe: outer FV discrete. ⇒ What about the inner FV ’s? Theorem If ϕ and ϕ1 denote Joe’s generators with ϑ ≤ ϑ1 , then ϕ−1 0,1 (t; v), v ∈ N, P∞ has Laplace Stieltjes inverse FV (x) = k=1 yk 1[xk ,∞) (x) with v X jϑ/ϑ1 j+k v xk = k and yk = , k ∈ N. (−1) j k j=0
• Precalculation numerically complicated: Runtime (dependence on v sample) and errors (FV (k) = 0, k < v and FV (v) = (ϑ/ϑ1 )v ). 14
Sampling Archimedean copulas
⇒ Not adequate for sampling purposes! Idea Use the generating function for the inner FV : g(x) = (1 − (1 − x)ϑ/ϑ1 )v .
Lemma (Devroye) Let g, g1 , g2 be g.f.’s such that g(t) = g1 (g2 (t)) and let N and X have g.f. g1 and g2 , respectively, then Y =
N X
Xi has generating function g,
i=1
where the Xi ’s are i.i.d. copies of X. 15
Sampling Archimedean copulas
Algorithm (1) Precalculate the outer FV once (yk =
1/ϑ k−1 (−1) , k
k ∈ N).
(2) Precalculate F corresponding to the g.f. 1 − (1 − x)ϑ/ϑ1 once ϑ/ϑ1 (yk = k (−1)k−1 , k ∈ N).
(3) Sample V ∼ FV .
(4) Sample i.i.d. Xi ∼ F , i ∈ {1, . . . , V } and build of the inner FV ).
PV
i=1 Xi
(a sample
(5) Proceed as before. • For Frank’s nested family: Similar. • For AMH’s nested family: Inner FV also a geometric distribution (no precalculation). 16
Sampling Archimedean copulas
• Scatterplot matrix for a fully nested Joe copula (ϑ = 1.2, ϑ1 = 1.6):
Component 3
Component 2
Component 1
17
Sampling Archimedean copulas
Mixed families • Is it possible to mix different families? • Examples from Nelson (1998): The sufficient nesting condition holds for 7 family combinations including (AMH,Clayton) for ϑ ∈ [0, 1), ϑ1 ∈ [1, ∞). A nested (AMH,Clayton) copula • Parameters ϑ = 0.8 and ϑ1 = 2. • Outer FV : Known geometric distribution. 1/ϑ1 + ϑ)−v ⇒ Inner F not • Inner FV : ϕ−1 V 0,1 (t; v) = ((1 − ϑ)(1 + t) explicitly known.
18
Sampling Archimedean copulas
Idea Apply numerical inversion of Laplace transforms and numerical rootfinding to obtain a realization from FV (x) = (L−1 (ϕ−1 (t)/t))(x), x ∈ [0, ∞). • Methods: – Fixed Talbot algorithm (contour deformation in the Bromwich integral). – Gaver Stehfest and Gaver Wynn rho algorithm (discrete analog of the Post-Widder formula). – Laguerre series algorithm (approximation via Laguerres series). • Comparison in the known nonnested Clayton case (ϑ = 0.8):
19
Sampling Archimedean copulas
Parameter calibration according to the precision requirement F (x) − F˜ (x) V V max < 0.0001 for P = {0.001, 0.002, . . . , 10}. x∈P FV (x)
for an approximation F˜V to FV . 2.4 Simulation results
• χ2 -test based on hypercubes of a partition into 5 parts for each dimension. • Maximal (MXDEV) and mean (MDEV) deviations of the probabilities of falling in the hypercubes. • Matrix of pairwise sample versions of Kendall’s tau. • Visual check of plots of outer and inner FV ’s. 20
Sampling Archimedean copulas
For the nested (AMH,Clayton) copula Based on 500.000 observations, ϑ = 0.8 and ϑ1 = 2, using the Fixed Talbot algorithm: • p-value = 1.000000000000 (χ2 -test). • MXDEV = 0.000555 and MDEV = 0.000084. • τˆϑ ∈ {0.2346, 0.2343} (τϑ = 0.2337) and τˆϑ1 = 0.5009 (τϑ1 = 0.5).
21
Sampling Archimedean copulas
1.0
1.0
• Plots for the outer (left) and inner (right) distribution functions FV :
(1) (2)
0.8 0.6
(4)
0.4 0.2
Value 0.4 0.2
Value
0.6
0.8
(3)
theta=0.8, theta2=2.0, v=0.1 theta=0.8, theta2=2.0, v=0.5 theta=0.8, theta2=2.0, v=1 theta=0.8, theta2=2.0, v=2
0.0
0.0
(1) (2) (3) (4)
0
5
10
15
20
0.0
Variable
0.2
0.4
0.6
Variable
22
0.8
1.0
Sampling Archimedean copulas
• Scatterplot matrix for this fully nested (AMH,Clayton) copula (ϑ = 0.8, ϑ1 = 2):
Component 3
Component 2
Component 1
23
Sampling Archimedean copulas
Runtimes For 500.000 observations of ϕ−1 [ϕ(u1 ) + ϕ(ϕ−1 1 [ϕ1 (u2 ) + ϕ1 (u3 )])], we obtain: • (AMH,Clayton): 37.67s. • Gumbel: 2.22s. Outer power Clayton: 2.71s. • Joe: 1.50s. Frank: 1.60s. AMH: 0.59s.
24
