5.2.1 Fully Nested Archimedean Copulas (FNAC)

For d-dimensional random variables modeled with FNAC, there are d – 1 bivariate copula functions, which result in dependence structure with partial exchangeability (Joe, 1997; Embrechts et al., 2003; Whelan, 2004; McNeil, 2007; Savu and Trede, 2010; among others). Figure 5.1 presents an example of a four-dimensional FNAC structure. The bivariate copula is the building block for FNAC. The FNAC structure is constructed, based on the degree of dependence between the pair variables, with the following procedures:

1. 
   

   i. Choose the variables with the highest degree of dependence (rank-based) as the first two variables (1 and 2).

   
   
2. 
   

   ii. Compute the empirical copula using variables 1 and 2.

   
   
3. 
   

   iii. Evaluate the degree of dependence (rank-based) between empirical copula from step ii with the remaining variables.

   
   
4. 
   

   iv. Choose variable 3, i.e., yielding the highest degree of dependence (rank-based) with the empirical copula built with variables 1 and 2.

   
   
5. 
   

   v. Continue the process until the last variable is considered.

Figure 5.1 Four-dimensional FNAC structure.

From Figure 5.1, it is seen that three bivariate copulas are needed to represent the dependence for the four-dimensional random variables through FNAC as follows. First, random variables u₁u1 and u₂u2 are coupled through copula C₃C3. Second, random variable u₃u3 is coupled with C₃(u₁, u₂)C3u1u2 through copula C₂C2. Third, random variable u₄u4 is coupled with C₂(u₃, C₃(u₁, u₂))C2u3C3u1u2 through copula C₁C1. Hence, a four-dimensional copula requires three bivariate copulas C₁, C₂C1,C2, and C₃C3, with generators ϕ₁ϕ1, ϕ₂ϕ2, and ϕ₃ϕ3 and may be written as follows:

C(u1,u2,u3,u4)=C1(u4,C2(u3,C3(u1,u2)))=ϕ1−1(ϕ1(u4)+ϕ1(ϕ2−1(ϕ2(u3)+ϕ2(ϕ3−1(ϕ3(u1)+ϕ3(u2))))))=ϕ1−1(ϕ1(u4)+ϕ1∘ϕ2−1(ϕ2(u3)+ϕ2∘ϕ3−1(ϕ3(u1)+ϕ3(u2))))(5.1)

where _○ represents the composition of functions.

Similarly, the FNAC for d-dimensional random variables (e.g., Joe, 1997; Embrechts et al., 2003; Whelan, 2004; Nelsen, 2006) may be generated as follows:

C(u1,…,ud)=ϕ1−1(ϕ1(ud)+ϕ1∘ϕ2−1(ϕ2(ud−1)+ϕ2∘…∘ϕd−1−1(ϕd−1(u1)+ϕd−1(u2))))(5.2)

It is worth noting that Equation (4.1) in Chapter 4, i.e., the exchangeable symmetric Archimedean copula, is a special case of Equation (5.2) if ϕ(θ₁) = ϕ₂(θ₂) = … = ϕ_d − 1(θ_d − 1) = ϕ(θ), θ₁ = θ₂ = … = θ_d − 1ϕθ1=ϕ2θ2=…=ϕd−1θd−1=ϕθ,θ1=θ2=…=θd−1. For the d-dimensional FNAC, the bivariate margins themselves are also Archimedean copulas that allow for free specification of d – 1 copulas with the remaining identified implicitly through FNAC (Whelan, 2004; Berg and Aas, 2007). Using Equation (5.1) (Figure 5.1) as an example, this statement may be expressed as follows: (i) there are three Archimedean copulas of free specification, i.e., C₃C3 with parameter θ₃θ3 for variables u₁, u₂u1,u2; C₂C2 with parameter θ₂θ2 for variables {u₃, C₃(u₁, u₂; θ₃}{u3,C3(u1,u2;θ3}; and C₁C1 with parameter θ₁θ1 for variables {u₄, C₂(u₃, C₃(u₁, u₂; θ₃); θ₂}{u4,C2(u3,C3u1u2θ3;θ2}; (ii) pairs (u₁, u₃), (u₂, u₃)u1u3,u2u3 have copula C₂C2 with parameter θ₂θ2; and (iii) pairs (u₁, u₄), (u₂, u₄), (u₃, u₄)u1u4,u2u4,u3u4 have copula C₃C3 with parameter θ₁θ1. The decreasing degree of dependence for the increasing levels of nesting (i.e., θ₁ ≤ θ₂ ≤ … ≤ θ_d − 1θ1≤θ2≤…≤θd−1 with θ₁θ1 and θ_d − 1θd−1 representing the parameters for the highest and lowest levels, respectively) is another technical condition for proper construction of the d-dimensional fully nested asymmetric Archimedean copula.

It should also be pointed out that the following conditions need to be satisfied for the nested generating functions:

• 
  

  ϕ1−1,ϕ2−1,…,ϕd−1−1 must satisfy the necessary conditions for being completely monotonic.

  
  
• 
  

  According to Embrechts et al. (2003), the coupling of functions wk=ϕk∘ϕk+1−1 belongs to a class of functions ℒ∞∗ defined as follows:

  
  
  ℒ∞∗=ω:0∞→0∞ω0=0ω∞=∞−1k−1dkωtdt≥0k=12…∞(5.3)
  
  

Based on Equation (5.2), the simplest three-dimensional FNAC (shown in Figure 5.2) can be written as follows:

Cu1u2u3=ϕ1−1(ϕ1u3+ϕ1∘ϕ2−1ϕ2u1+ϕ2u2(5.4)

Figure 5.2 Three-dimensional FNAC structure.

In accordance with Equation (5.4), we outline here the derivation of five three-dimensional asymmetric Archimedean copulas that are commonly applied.

M3 (Joe, 1997):

C2u1u2=−1θ2ln1−1−e−θ2u11−e−θ2u21−e−θ2

Let t = C₂(u₁, u₂)t=C2u1u2. Then we have C1u3t=−1θ1ln1−1−e−θ1u3(1−e−θ1t1−e−θ1

Cu1u2u3=C1u3C2u1u2=C1u3t=−1θ1ln1−1−e−θ1u31−1−e−θ2u11−e−θ2u21−e−θ21−e−θ1(5.5)

θ₂ ≥ θ₁ ∈ [0, ∞), τ₁₂, τ₁₃, τ₂₃ ∈ [0, 1]θ2≥θ1∈0∞,τ12,τ13,τ23∈01 for positive dependent trivariate variables.

The M3 copula may be also called the asymmetric trivariate Frank copula.

We now use the following specific examples to illustrate these marginal distributions.

Example 5.1 Derive the M3 copula for θ₁ = 2.0 and θ₂ = 3.0θ1=2.0andθ2=3.0 by setting u₃ = 0.6u3=0.6. Assuming u₁~F₁(x₁) : X₁~gamma(2, 4); u₂~F₂(x₂) : X₂~normal(1, 3²); u₃~F₃(x₃) : X₃~EV1(10, 7)u1~F1x1:X1~gamma24;u2~F2x2:X2~normal132;u3~F3x3:X3~EV1107, and {X₁, X₂}X1X2 has a higher pairwise dependence.

Solution: With {X₁, X₂}X1X2 having higher pairwise dependence, we first couple X₁and X₂X1andX2 and build the copula function from the marginals as follows:

u1=F1x1=1Γ2γ4×1γ:incompletegammafunction

u2=F2x2=Φx2−13,Φ:Standardnormaldistribution

u3=F3x3=exp−exp−x3−107

Since we already set u₃ = 0.6u3=0.6, then we have x₃≈9.388x3≈9.388 from the EV1 population.

Finally, we can write the fully nested copula using the M3 copula as follows:

C2u1u23=−13.0ln1−1−e−3.0u11−e−3.0u21−e−3.0=−13.0ln1−(1−e−3.0(1Γ2γ4×1))1−e−3.0Φx2−131−e−3.0

Cu1u20.632=C0.6C2u1u232=−12.0ln1−1−e−2.00.61−1−e−3.0u11−e−3.0u21−e−3.01−e−2.0=−12.0ln1−1−e−2.00.61−(1−e−3.0(1Γ2γ4×1))1−e−3.0Φx2−131−e−3.01−e−2.0

Figure 5.3(a) plots the corresponding joint CDF for the derived M3 copula with u₃ = 0.6u3=0.6. M4 (Joe, 1997):

C2u1u2=u1−θ2+u2−θ2−1−1θ2

Let t = C₂(u₁, u₂)t=C2u1u2. Then we have C1u3t=u3−θ1+t−θ1−1−1θ1

Cu1u2u3=C1u3C2u1u2=u1−θ2+u2−θ2−1−θ1θ2+u3−θ1−1−1θ1(5.6)

θ₂ ≥ θ₁ ∈ [0, ∞), τ₁₂, τ₁₃, τ₂₃ ∈ [0, 1]θ2≥θ1∈0∞,τ12,τ13,τ23∈01 for positive dependent trivariate variables. The M4 copula may also be called the trivariate asymmetric Clayton copula.

Figure 5.3 Joint CDF for derived FNACs: (a) M3 copula, (b) M4 copula, (c) M5 copula, (d) M6 copula, and (e) M12 copula.

Example 5.2 Derive the M4 copula using information given in Example 5.1.

Solution: In Example 5.1, we have θ₁ = 2.0, θ₂ = 3.0θ1=2.0,θ2=3.0 by setting u₃ = 0.6u3=0.6. Thus, we have the following:

C2u1u23=u1−3.0+u2−3.0−1−13.0=1Γ2γ4×1−3.0+Φx2−13−3.0−1−13.0

Cu1u20.632=C1C2u1u230.6=u1−3.0+u2−3.0−123+0.6−2.0−1−12.0=1Γ2γ4×1−3.0+Φx2−13−3.0−123+0.6−2.0−1−12.0

Figure 5.3(b) plots the corresponding joint CDF for the derived M4 copula with u₃ = 0.6u3=0.6.

M5 (Joe, 1997):

C2u1u2=1−1−u1θ2+1−u2θ2−1−u1θ21−u2θ21θ2

Let t=C2u1u2,1−t=1−u1θ2+1−u2θ2−1−u1θ21−u2θ21θ2. Then we have the following:

Cu1u2u3=C1u3C2u1u2=1−1−u1θ21−1−u2θ2+1−u2θ2θ1θ21−1−u3θ1+1−u3θ11θ1(5.7)

θ₂ ≥ θ₁ ∈ [1, ∞), τ₁₂, τ₁₃, τ₂₃ ∈ [0, 1]θ2≥θ1∈1∞,τ12,τ13,τ23∈01. The M5 copula may also be called the trivariate asymmetric Joe copula.

Example 5.3 Derive M5 copula using the information given in Example 5.1.

Solution: In Example 5.1, we have θ₁ = 2.0, θ₂ = 3.0θ1=2.0,θ2=3.0 by setting u₃ = 0.6u3=0.6. Thus we have the following:

C2u1u23.0=1−1−u13.0+1−u23.0−1−u13.01−u23.013.0

=1−(1−1Γ2γ4×13.0+1−Φx2−133.0−1−1Γ2γ4×11−Φx2−133.0)13.0

Cu1u20.632=1−(1−u13.01−1−u23.0+1−u23.02.03.01−0.42.0+0.42.0)12.0

Figure 5.3(c) plots the corresponding joint CDF for the derived M5 copula with u₃ = 0.6u3=0.6.

M6 (Joe, 1997; Embrechts, 2003):

Let C2u1u2=e−−lnu1θ2+−lnu2θ21θ2,and

t=C2u1u2,−lnt=(−lnu1)θ2+−lnu2θ21θ2.Thenwehave

Cu1u2u3=C1(u3,C2u1u2=e−−lnu1θ2+−lnu2θ2θ1θ2+−lnu3θ11θ1(5.8)

θ₂ ≥ θ₁ ∈ [1, ∞), τ₁₂, τ₁₃, τ₂₃ ∈ [0, 1]θ2≥θ1∈1∞,τ12,τ13,τ23∈01 for positive dependent trivariate variables. The M6 copula may also be called the trivariate asymmetric Gumbel–Hougaard copula.

Example 5.4 Derive the M6 copula using the information given in Example 5.1.

Solution: In Example 5.1, we have θ₁ = 2.0, θ₂ = 3.0θ1=2.0,θ2=3.0 by setting u₃ = 0.6u3=0.6. Thus we have the following:

C2u1u23=e−−lnu13.0+−lnu23.013.0

Cu1u20.632=e−−lnu13.0+−lnu23.023+−ln0.62.012.0

Figure 5.3(d) plots the corresponding joint CDF for the derived M6 copula with u₃ = 0.6u3=0.6.

M12 (Embrechts, 2003):

C2u1u2=11+1u1−1θ2+1u2−1θ21θ2

Let t=C2u1u2,1t−1=1u1−1θ2+1u2−1θ21θ2. Then we have

Cu1u2u3=11+1u1−1θ2+1u2−1θ2θ1θ2+1u3−1θ11/θ1(5.9)

θ2≥θ1∈1∞,τ12,τ13,τ23∈131.

Example 5.5 Derive the M12 copula using the information given in Example 5.1.

Solution:

C2u1u23=11+1u1−13.0+1u2−13.013.0

Cu1u20.632=11+1u1−13.0+1u2−13.023+10.6−12.01/2.0

Figure 5.3(e) plots the joint CDF for the derived M12 copula with u₃ = 0.6u3=0.6.

Example 5.6 Derive a four-dimensional FNAC copula function based on the bivariate Frank copula.

Solution: From Figure 5.1, we have the following:

C3u1u2θ3=−1θ3ln1−1−e−θ3u11−e−θ3u21−e−θ3

C₃C3 and u₃u3 are coupled as copula C₂(C₃, u₃)C2C3u3 with parameter θ₂θ2, which can be written as follows:

C2u1u2u3=C2C3u3θ2=−1θ2ln1−1−1−1−e−θ3u11−e−θ3u21−e−θ3θ2θ31−e−θ2u31−e−θ2

Finally, C₂C2 and u₄u4 are defined as copula C₁(C₂, u₄)C1C2u4 with parameter θ₁θ1, which results in C₁(C₂, u₄; θ₁) = C(u₁, u₂, u₃, u₄; θ₁, θ₂, θ₃)C1C2u4θ1=Cu1u2u3u4θ1θ2θ3 as follows:

Cu1u2u3u4θ1θ2θ3=−1θ1ln1−1−e−θ1u41−e−θ11−1−1−1−1−e−θ3u11−e−θ3u21−e−θ3θ2θ31−e−θ2u31−e−θ2θ1θ2

In the same way as for the previous examples, for the four-dimensional random variables {X_i, i = 1, …, 4}Xii=1…4, the random variable X_iXi may follow different marginal distributions as follows:

u₁ = F₁(x₁); u₂ = F₂(x₂); u₃ = F₃(x₃); u₄ = F₄(x₄)

u1=F1x1;u2=F2x2;u3=F3x3;u4=F4x4.Asanillustration,wecansay,

X₁~ exp (λ₁) ⇒ u₁ = F₁(x₁) = 1 −  exp (−λ₁x₁);

X1~expλ1⇒u1=F1x1=1−exp−λ1×1;

X2~gammaαβ⇒u2=F2x2=1Γαγβx2;

X3~logisticab⇒u3=F3x3=11+expx−ab;

X4~PearsonIIIcαβ⇒u4=F4x4=1Γαγβx−c.

5.2.2 Partially Nested Archimedean Copulas (PNAC)

Originally, Joe (1997) proposed the structure of PNAC as an alternative approach for FNAC. PNAC may be considered a composite of EAC and FNAC (Berg and Aas, 2007).Similar to FNAC, PNAC also has d – 1 bivariate copulas that are partially exchangeable. As a simple example, Figure 5.4 illustrates the PNAC structure for four-dimensional random variables: (1) couple the two pairs (u₁, u₂)u1u2 and (u₃, u₄)u3u4 with copula C₃C3 with parameter θ₃θ3 and C₂C2 with parameter θ₂,θ2, respectively, at the first level; and (2) the third copula C₁C1 with parameter θ₁θ1 will be applied to couple C₂C2 and C₃C3 at the second level (Berg and Aas, 2007). Figure 5.4 also shows (1) exchangeability between u₁u1 and u₂u2, as well as between u₃u3 and u₄u4; and (2) four pairs (u₁, u₃), (u₁, u₄), (u₂, u₃)u1u3,u1u4,u2u3, and (u₂, u₄)u2u4 all have copula C₁C1. Furthermore, the same constraints on parameters for FNAC are required to be satisfied for PNAC (Berg and Aas, 2007), i.e., (i) PNAC may be used to model the positively dependent variables, and (ii) the dependence decreases with the increase of nesting levels (i.e., the parameters of a higher level are smaller than those of a lower level).

Figure 5.4 Partially nested Archimedean construction.

Example 5.7 Using the bivariate Frank copula as the building block to derive a four-dimensional PNAC function for the structure given in Figure 5.4.

Solution: As shown in Figure 5.4, (u₁, u₂)u1u2 and (u₃, u₄)u3u4 can be represented through the Frank copula as follows:

C3u1u2θ3=−1θ3ln1−1−e−θ3u11−e−θ3u2e−θ3

C4u3u4θ2=−1θ2ln1−1−e−θ2u31−e−θ2u4e−θ2

Then C₁C1 can be represented through C₃, C₂C3,C2 as follows:

Cu1u2u3u4θ1θ2θ3=C1C3C2θ1=−1θ1ln1−1−e−θ1C31−e−θ1C21−e−θ1=−1θ1ln1−1−1−1−e−θ3u11−e−θ3u21−e−θ3θ1θ31−1−1−e−θ2u31−e−θ2u41−e−θ2θ1θ21−e−θ1

with the parameters: 0 ≤ θ₁ ≤ θ₂, θ₃0≤θ1≤θ2,θ3.

In the same manner for FNAC, random variables {X₁ : i = 1, 2, 3, 4}X1:i=1234 may follow different marginal distributions as u_i = F_i(x_i)ui=Fixi.

5.2.3 General Case

Originating in Joe (1997), the general nested Archimedean copula (GNAC) construction was further developed by Whelan (2004) and Savu and Trede (2006). Savu and Trede (2006) first introduced the notation for arbitrary nesting and the procedure for calculating the d-dimensional probability density function in general. To build a hierarchy of Archimedean copulas, they also applied the notation for the hierarchical Archimedean copula for GNAC. The main idea of the generally nested Archimedean construction is presented in this section (Berg and Aas, 2007).

For the GNAC with L levels, there are n_lnl distinct objects (an object is either a copula or a variable) at each level l. At level l = 1l=1, variables u₁, …, u_du1,…,ud are grouped into n₁n1 exchangeable multivariate Archimedean copulas. These copulas are, in turn, coupled with n₂n2 copula at level l = 2l=2, and so on. Berg and Aas (2007) presented an example of a nine-dimensional copula to explain this structure (Figure 5.5).

Figure 5.5 Hierarchically nested Archimedean copula construction.

Following Figure 5.5, the nine-dimensional copula can be written as

At the first level, there are two two-dimensional EACs, i.e., C₄₁(u₁, u₂)C41u1u2 with parameter θ₄₁θ41 and C₄₂(u₈, u₉)C42u8u9 with parameter θ₄₂θ42. There are one three-dimensional and one two-dimensional EACs at the second level, i.e., C₃₁(C₄₁, u₃, u₄)C31C41u3u4 with parameter θ₃₁θ31 and C₃₂(u₇, C₄₂)C32u7C42 with parameter θ₃₂θ32. At the third level, there is only one copula, C₂₁(C₃₁, u₅, u₆)C21C31u5u6 with parameter θ₂₁θ21. At the top (fourth) level, the copula C₁₁C11, with parameter θ₁₁θ11, is applied to model the dependence between C₂₁C21 and C₃₂C32.

To ensure that GNAC is a valid Archimedean copula, there are a number of conditions that need to be satisfied (Savu and Trede, 2006; Berg and Aas, 2007):

1. 
   

   a. The number of copulas must decrease with the increasing level of nesting. The top level may contain only one copula, and the inverse of the generating functions (ϕ⁻¹ϕ−1) must be completely monotonic.

   
   
2. 
   

   b. The dependence of GNAC must decrease with the increasing level of nesting. For example, in Figure 5.5, parameters must be stratified following the condition θ₄₁ ≥ θ₃₂ ≥ θ₂₁ ≥ θ₁₁θ41≥θ32≥θ21≥θ11 and θ₄₂ ≥ θ₃₂ ≥ θ₁₁θ42≥θ32≥θ11. However, when mixing copula generators that belong to different Archimedean copula families, this requirement might not be sufficient. Two Archimedean copulas from different families (i.e., Fam1 and Fam2) can only be nested if the derivative of the product ϕ1∘ϕ2−1 is completely monotonic. Joe (1997) presented details about copula families that can be mixed and explored structures where all the generators are from the same family are explored, and the other structures are still not fully explored.

5.2.4 Parameter Estimation for Nested Copulas

For NAC with an explicit density expression, the maximum likelihood estimation method is commonly applied to estimate the copula parameters; however, the NAC density function may not be straightforwardly derived. Savu and Trede (2006) proposed a recursive approach to derive the density function for general NAC. With this approach, the number of computational steps for evaluating the density increases rapidly with the copula complexity, and parameter estimation becomes very time consuming in higher dimensions (Savu and Trede, 2006; Berg and Aas, 2007).

The density function of NAC can be derived using the chain rule as discussed by Savu and Trede (2006). We will use the following examples to illustrate the general procedure on how to apply the chain rule. Furthermore, we derive the density functions for the M3, M4, M5, M6, and M12 copulas (Joe, 1997) in the appendix as specific examples.

Example 5.8 Derive the density function for three-dimensional FNAC (Equation (5.4) corresponding to Figure 5.2).

Solution: Equation (5.4) may be rewritten as follows:

C(u₁, u₂, u₃) = C₁(C₂(u₁, u₂), u₃)Cu1u2u3=C1C2u1u2u3 and its density, i.e., c(u₁, u₂, u₃)cu1u2u3, may be derived as follows:

∂Cu1u2u3∂u1=∂C1C2u1u2u3∂u1=∂C1∂C2∂C2∂u1;∂C2u1u2.u3∂u1∂u2=∂2C1∂C22∂C2∂u2∂C2∂u1+∂2C1∂C2∂2C2∂u1∂u2

Finally, we have the following:

cu1u2u3=∂3Cu1u2u3∂u1∂u2∂u3=∂3C1∂C22∂u3∂C2∂u2∂C2∂u1+∂2C1∂C2∂u3∂2C2∂u1∂u2

Example 5.9 Derive the density function for four-dimensional FNAC (i.e., Equation (5.1) corresponding to Figure 5.1).

Solution: Following Equation (5.1) and Figure 5.1, we have the following:

C(u₁, u₂, u₃, u₄) = C₁(u₄, C₂) = C₁(u₄, C₂(u₃, C₃(u₁, u₂)))Cu1u2u3u4=C1u4C2=C1u4C2u3C3u1u2 and its density c(u₁, u₂, u₃, u₄)cu1u2u3u4 may be derived as follows:

∂Cu1u2u3u4∂u1=∂C1u4C2u3C3u1u2∂u1=∂C1∂C2∂C2∂C3∂C3∂u1∂2Cu1u2u3u4∂u1∂u2=∂2C1∂C22∂C2∂C32∂C3∂u1∂C3∂u2+∂C1∂C2∂2C2∂C32∂C3∂u1∂C3∂u2+∂C1∂C2∂C2∂C3∂2C3∂u1∂u2∂3Cu1u2u3u4∂u1∂u2∂u3=∂3C1∂C22∂C2∂C32∂C2∂u3∂C3∂u1∂C3∂u2+2∂2C1∂C22∂C2∂C3∂2C2∂C3∂u3∂C3∂u1∂C3∂u2=∂2C1∂C22∂C2∂u3∂2C2∂C32∂C3∂u1∂C3∂u2=∂C1∂C2∂3C2∂C32∂u3∂C3∂u1∂C3∂u2+∂2C1∂C22∂C2∂u3∂C2∂C3∂2C3∂u1∂u2+∂C1∂C2∂2C2∂C3∂u3∂2C3∂u1∂u2

Finally, we have the following:

cu1u2u3u4=∂4Cu1u2u3u4∂u1∂u2∂u3∂u4=∂4C1∂C23∂u4∂C2∂C32∂C2∂u3∂C3∂u1∂C3∂u2+2∂3C1∂C22∂u4∂C2∂C3∂2C2∂C3∂u3∂C3∂u1∂C3∂u2+∂3C1∂C22∂u4∂C2∂u3∂2C2∂C32∂C3∂u1∂C3∂u2+∂2C1∂C2∂u4∂3C2∂C32∂u3∂C3∂u1∂C3∂u2+∂3C1∂C22∂u4∂C2∂u3∂C2∂C3∂2C3∂u1∂u2+∂2C1∂C2∂u4∂2C2∂C3∂u3∂2C3∂u1∂u2

Example 5.10 Derive the density function for the copula function represented by Figure 5.4.

Solution: According to Figure 5.4, we have the following: C(u₁, u₂, u₃, u₄) = C₁(C₃(u₁, u₂), C₂(u₃, u₄)).Cu1u2u3u4=C1C3u1u2C2u3u4. Then its density function c(u₁, u₂, u₃, u₄)cu1u2u3u4 may be expressed as follows:

∂Cu1u2u3u4=∂C1∂C3∂C3∂u1

∂2Cu1u2u3u4∂u1∂u2=∂2C1∂C32∂C3∂u2∂C3∂u1+∂C1∂C3∂2C3∂u1∂u2

∂3Cu1u2u3u4∂u1∂u2∂u3=∂3C1∂C32∂C2∂C2∂u3∂C3∂u2∂C3∂u1+∂2C1∂C3∂C2∂C2∂u3∂2C3∂u1∂u2

Finally, we have the following:

cu1u2u3u4=∂4Cu1u2u3u4∂u1∂u2∂u3∂u4=∂4C1∂C32∂C22∂C2∂u4∂C2∂u3∂C3∂u2∂C3∂u1+∂3C1∂C32∂C2∂2C2∂u3∂u4∂C3∂u2∂C3∂u1+∂3C1∂C3∂C22∂C2∂u4∂C2∂u3∂2C3∂u1∂u2+∂2C1∂C3∂C2∂2C2∂u3∂u4∂2C3∂u1∂u2

With the copula density function derived, we can then apply MLE to estimate parameters simultaneously with the constraints of parameters at a lower level being larger than those at a higher level. However, the copula parameters may also be estimated sequentially with the use of MLE as follows:

1. 
   

   i. Estimate the copula parameter at the lowest level.

   
   
2. 
   

   ii. Estimate the copula parameter for the second-lowest level by fixing the parameters estimated for the lowest level.

   
   
3. 
   

   iii. Repeat the preceding steps until we reach the top level of the NAC structure.

5.2.5 Simulation for Nested Copulas

In the previous chapters, we have shown that EAC may be simulated with several methods, such as Laplace transform (LT) and CPI Rosenblatt’s transform, and through its unique generating function ϕϕ with a simple algorithm. Frees and Valdez (1998) showed how to use the LT method to simulate NACs for the generators taken from either the Gumbel– Hougaard or the Clayton copula family. However, Berg and Aas (2007) have pointed out that the LT method is limited to the copulas such that we can find a distribution that equals the LT of the inverse generating function and from which we can easily sample. In most cases, the LT method needs to obtain the d – 1 first derivatives of the copula function, which usually yield extremely complex expressions under higher-order derivatives. The limitation of LT method may cause the simulation to become inefficient for high dimensions (Berg and Aas, 2007).

Compared to the LT method, the CPI Rosenblatt transform method is more universal and will be introduced to simulate from NAC. Let X = {X₁, X₂, …, X_d}X=X1X2…Xd be a d-dimensional random vector with marginal distributions F(x_i)Fxi and conditional distributions F(x_i| x₁, …, x_i − 1), i = 1, …, dFxix1…xi−1,i=1,…,d. The CPI Rosenblatt’s transform of X is defined as T(X) = {T(X₁), …, T(X_d)}TX=TX1…TXd:

With the use of CPI method, random variables are simulated with the following procedure:

1. 
   

   i. Generate W = {w₁, w₂, …, w_d}W=w1w2…wd independent random variables following the uniform distribution [0, 1].

   
   
2. 
   

   ii. Set x₁ = w₁x1=w1.

   
   
3. 
   

   iii. Set w₂ = T(X₂) = F_2 ∣ 1(x₂| x₁)w2=TX2=F2∣1x2x1 to obtain x2=F2∣1−1w2x1.

   
   
4. 
   

   iv. Set w₃ = T(X₃) = F_{3 ∣ 1, 2}(w₃| x₁, x₂)w3=TX3=F3∣1,2w3x1x2 to obtain x3=F3∣1,2−1w3x1x2.

   
   

   …

   
   

   Set w_d = T(X_d) = F_{d ∣ 1, 2, …d − 1}(w_d| x₁, x₂, …, x_d)wd=TXd=Fd∣1,2,…d−1wdx1x2…xd.

Example 5.11 Assuming the pseudo-observations given in Table 5.1 may be modeled with the M6 copula, (1) estimate the copula parameters both simultaneously and sequentially using MLE; and (2) simulate the random variables with a sample size of 50.

Table 5.1. Trivariate pseudo-observations.

	u1u1	u2u2	u3u3
1	0.241	0.138	0.103
2	0.241	0.172	0.172
3	0.241	0.241	0.276
4	0.241	0.586	0.655
5	0.793	0.828	0.897
6	0.483	0.345	0.379
7	0.931	0.914	0.621
8	0.724	0.759	0.724
9	0.414	0.621	0.586
10	0.759	0.414	0.310
11	0.862	0.793	0.793
12	0.655	0.517	0.448
13	0.414	0.379	0.552
14	0.569	0.448	0.414
15	0.569	0.690	0.690
16	0.414	0.310	0.241
17	0.241	0.552	0.862
18	0.069	0.035	0.035
19	0.241	0.276	0.345
20	0.069	0.069	0.069
21	0.897	0.914	0.931
22	0.655	0.655	0.483
23	0.069	0.103	0.138
24	0.241	0.207	0.207
25	0.655	0.724	0.759
26	0.517	0.483	0.517
27	0.828	0.862	0.828
28	0.966	0.966	0.966

Solution: Estimate the copula parameters.

To estimate the parameters for the fitted M6 copula, we use Figure 5.2 as the FNAC scheme.

• 
  

  Estimate the copula parameters simultaneously.

  
  

  To estimate the copula parameters simultaneously, the copula density function (i.e., Equation (M6–3) in the appendix) is applied to write the log-likelihood function as follows:

  
  
  logL=∑1u1u2u3−lnu1θ2−1−lnu2θ2−1−lnu3θ3−1e−w1θ1(G2θ1θ2−2w3θ1−3+2θ1−2w2θ1−2+θ2−θ1Gθ1θ2−2w2θ1−2+θ1−12θ1−1G2θ1θ2−2w1θ1−3+θ1−1G2θ1θ2−2w2θ1−3+θ1−1θ2−θ1Gθ1θ2−2w1θ1−2)
  where G =−lnu1θ2+−lnu2θ2;w=−lnu3θ1+[−lnu1θ2+−lnu2)θ1θ1θ2.
  
  
  

  The parameter constraint is given as 1 ≤ θ₁ ≤ θ₂1≤θ1≤θ2, where θ₂θ2 corresponds to the parameters for the first level.

  
  

  Maximizing the log-likelihood function numerically (e.g., using genetic algorithm ga function in MATLAB), the parameters are estimated as follows:

  
  
  
  

  θ₂ = 4.4158; θ₁ = 3.3532

  θ2=4.4158;θ1=3.3532.
  
  
  
  

  It is worth noting that to properly estimate the parameters simultaneously, the linear constraint needs to be applied with vector A = [–1,1] B = 0, which represents –θ₂ + θ₁ ≤ 0–θ2+θ1≤0.

  
  
• 
  

  Estimate the copula parameters sequentially.

  
  

  To estimate the copula parameters sequentially, the density function for the bivariate Gumbel–Hougaard copula is applied (Chapter 4).

  
  
  
  
  • 
    

    Step 1: Maximizing the log-likelihood function for (u₁, u₂)u1u2, we have θ₂ = 4.4682θ2=4.4682.

    
    
  • 
    

    Step 2: Compute C(u₁, u₂; θ₂ = 4.4682C(u1,u2;θ2=4.4682) and estimate the parameter for

    
    
  • 
    

    (u₃, C(u₁, u₂; θ₂ = 4.4682))u3Cu1u2θ2=4.4682. Again using MLE, we have θ₁ = 3.2088θ1=3.2088. It is worth noting that to estimate the parameter (i.e., the Gumbel–Hougaard copula) for the top level, the lower and upper bounds are [1, θ₂]1θ2.

    
    
  • 
    

    Finally, for both simultaneous and sequential estimation, the parameters estimated are coded as follows:

    
    
  • 
    

    param = [param(1), param(2)] = [θ₂, θ₁]param=param1param2=θ2θ1; param(1) and param(2) represents bottom and top levels, respectively.

  
  
  
  
• 
  

  Simulation from the fitted M6 copula.

  
  

  As discussed previously, the random variates are simulated using the CPI Rosenblatt transform, as shown in Figure 5.6(a).

In addition, we have discussed previously that [u₁, u₃]u1u3 and [u₂, u₃]u2u3 may be modeled with the Gumbel–Hougaard copula with parameter θ₁θ1. Figure 5.6(b) compares the simulation as well as the box plot of simulated and sample Kendall’s tau (100 simulations with a sample size of 28).

Figure 5.6 (a) Comparison of pseudo-observations with those simulated from M6 copula; (b) simulation comparison from the Gumbel–Hougaard copula with parameter θ₁θ1 for (u₁, u₃), (u₂, u₃)u1u3,u2u3 directly; (c) comparison of sample Kendall’s tau with simulated Kendall’s tau from Gumbel–Hougaard copula with parameter θ = 2.8816θ=2.8816.

Example 5.12 Assuming the Gumbel–Hougaard copula may be applied as a biviarate building block, and using the scheme shown in Figure 5.4 and the pseudo-observations listed in Table 5.2, (1) estimate the copula parameters; and (2) simulate random variates with fitted copula for a sample size of 100.

Table 5.2. Pseudo-observations for Example 5.12.

	u1u1	u2u2	u3u3	u4u4
1	0.194	0.338	0.421	0.545
2	0.819	0.901	0.743	0.705
3	0.614	0.639	0.615	0.662
4	0.235	0.208	0.298	0.292
5	0.792	0.755	0.865	0.894
6	0.433	0.517	0.559	0.480
7	0.130	0.197	0.095	0.087
8	0.570	0.583	0.802	0.680
9	0.128	0.274	0.256	0.137
10	0.218	0.116	0.262	0.481
11	0.468	0.367	0.367	0.439
12	0.490	0.434	0.391	0.515
13	0.194	0.083	0.019	0.042
14	0.120	0.227	0.178	0.289
15	0.676	0.601	0.759	0.673
16	0.990	0.990	0.991	0.993
17	0.657	0.777	0.942	0.950
18	0.226	0.174	0.284	0.134
19	0.828	0.857	0.836	0.916
20	0.373	0.367	0.151	0.249
21	0.698	0.656	0.727	0.584
22	0.645	0.738	0.641	0.787
23	0.025	0.051	0.034	0.199
24	0.298	0.300	0.470	0.394
25	0.906	0.936	0.955	0.950
26	0.658	0.476	0.556	0.647
27	0.302	0.158	0.224	0.105
28	0.581	0.393	0.733	0.779
29	0.371	0.433	0.179	0.145
30	0.169	0.537	0.213	0.344
31	0.041	0.083	0.009	0.059
32	0.982	0.978	0.928	0.935
33	0.585	0.162	0.326	0.312
34	0.618	0.753	0.661	0.633
35	0.280	0.622	0.400	0.574
36	0.902	0.969	0.879	0.904
37	0.440	0.648	0.587	0.811
38	0.243	0.147	0.281	0.524
39	0.044	0.081	0.177	0.052
40	0.122	0.149	0.229	0.180
41	0.497	0.645	0.528	0.545
42	0.701	0.644	0.745	0.599
43	0.323	0.538	0.806	0.796
44	0.013	0.044	0.063	0.041
45	0.651	0.721	0.774	0.646
46	0.190	0.298	0.773	0.841
47	0.520	0.772	0.636	0.542
48	0.926	0.943	0.900	0.812
49	0.468	0.447	0.518	0.633
50	0.868	0.894	0.893	0.905
51	0.422	0.710	0.727	0.560
52	0.888	0.835	0.868	0.823
53	0.372	0.590	0.734	0.792
54	0.132	0.116	0.095	0.041
55	0.429	0.288	0.219	0.125
56	0.390	0.366	0.375	0.172
57	0.983	0.986	0.991	0.990
58	0.980	0.988	0.976	0.974
59	0.308	0.318	0.147	0.193
60	0.932	0.913	0.943	0.933

Solution:

1. 
   

   1. Estimate the copula parameters.

   
   

   According to Figure 5.4, let us use θ₁₂, θ₃₄θ12,θ34 to represent the copula parameters of [u₁, u₂], [u₃, u₄]u1u2,u3u4 at the bottom level and θθ to represent the copula parameter at the top level.

   
   
   
   
   • 
     

     Estimate the parameters simultaneously.

     
     

     Given the Gumbel–Hougaard copula as a bivariate building block, the copula density function for the four-dimensional PNAC Gumbel–Hougaard copula may be derived based on the chain rule following the procedure given in Example 5.10. With the parameter constraints 1 ≤ θ ≤ θ₁₂, θ₃₄1≤θ≤θ12,θ34, i.e.,

     
     

     θ−θ12≤0θ−θ34≤0, the inequality vector is then given as A=−1,0,10,−1,1,B=00, with the parameter set as param = [θ₁₂, θ₃₄, θ]param=θ12θ34θ.

     
     

     The parameters can be estimated numerically by maximizing the log-likelihood function with the preceding linear constraint as follows:

     
     
     
     

     θ₁₂ = 3.6949, θ₃₄ = 4.5035, θ = 2.8816

     θ12=3.6949,θ34=4.5035,θ=2.8816.
     
     
     
     
   • 
     

     Estimate the parameters sequentially.

     
     

     With the same estimation procedures shown in Example 5.11:

     
     

     The parameter for (u₁, u₂)u1u2 is estimated as θ₁₂ = 3.8545θ12=3.8545.

     
     

     The parameter for (u₃, u₄)u3u4 is estimated as θ₃₄ = 4.3949θ34=4.3949.

     
     

     The parameter for {C₃(u₁, u₂; θ₁₂), C₂(u₃, u₄; θ₃₄)}C3u1u2θ12C2u3u4θ34 is estimated by fixing θ₁₂, θ₃₄θ12,θ34 as θ = 3.3297θ=3.3297.

   
   
2. 
   

   2. Simulate random variates.

   
   

   Using the CPI Rosenblatt transform, Figure 5.7(a) compares the pseudo-observations in Table 5.2 with those simulated from the fitted PNAC Gumbel–Hougaard copula function.

Figure 5.7 (a) Comparison of pseudo-observations with those simulated with the parameters estimated simultaneously (θ₁₂ = 3.6949, θ₃₄ = 4.5035, θ = 2.8816θ12=3.6949,θ34=4.5035,θ=2.8816); (b) comparison of observed variables with simulated variables with θ = 2.8816θ=2.8816; (c) comparison of sample Kendall’s tau with the simulated Kendall’s taus.

As discussed previously for the PNAC structure, we know (u₁, u₃), (u₁, u₄), (u₂, u₃), (u₂, u₄)u1u3,u1u4,u2u3,u2u4 should have the same joint distribution that may be modeled using the Gumbel–Hougaard copula with parameter at the top level, i.e., θ = 2.8816θ=2.8816 with the comparison of simulated random variable and Kendall’s tau as shown in Figure 5.7(b) and 5.7(c). Figure 5.7(b) and 5.7(c) indicates that the preceding four pairs may be modeled using the same Gumbel–Hougaard copula.

5.3.1 Principle of Pair-Copula Decomposition of General Multivariate Distribution

Following Aas et al. (2009), we introduce the pair-copula decomposition of general multivariate distributions.

Let X = (X₁, X₂, …, X_d)X=X1X2…Xd be a vector of random variables with a joint density function f(x₁, …, x_d)fx1…xd. According to the conditional probability theory, the joint density function can be defined as follows:

In Chapters 3 and 4, the multivariate distribution F with marginals F₁(x₁), …, F_d(x_d)F1x1,…,Fdxd is defined using Sklar’s theorem as follows:

Fx1…xd=CF1x1…FdxdorCu1…ud=FF1−1×1…Fd−1xd(5.13)

where u_i = F_i(x_i)ui=Fixi; Fi−1ui is the inverse distribution of marginal F_i(x_i)Fixi.

Then, for an absolutely continuous F with strictly increasing, continuous marginal probability densities f₁(x₁), …, f_d(x_d)f1x1,…,fdxd, applying ∂d∂x1…∂xd to Equation (5.13), we have

fx1…xd=∂d∂F1x1…∂FdxdCF1x1…Fdxd∂F1x1∂x1…∂Fdxd∂xd(5.14a)

fx1…xd=c1,2,…,dF1x1…Fdxd∏i=1dfixi(5.14b)

where c_{1, 2, …, d}(⋅)c1,2,…,d⋅ stands for the d-dimensional copula density function.

In the bivariate case, Equation (5.14b) can be simplified to

where c₁₂(⋅)c12⋅ is the appropriate pair-copula density.

Using the conditional probability in Equation (5.12), the conditional probability density function can be easily written as

fx1x2=fx1x2f2x2=c12F1x1F2x2f1x1f2x2f2x2=c12F1x1F2x2f1x1(5.16)

Likewise, we have

Similarly, in the trivariate case, we can obtain the conditional probability density function:

fx1x2x3=fx1x2x3fx2x3=fx3fx1x2x3fx3fx2x3=fx1x2x3fx2x3(5.18)

According to the definition of conditional copula, we have

fx1x2x3=∂2Fx1x2x3∂x1∂x2=∂2∂x1∂x2C12∣3F1∣3x1x3F2∣3x2x3=∂2C12∣3F1∣3x1x3F2∣3x2x3∂F1∣3x1x3∂F2∣3x2x3∂F1∣3x1x3∂x1∂F2∣3x2x3∂x2=c12∣3F1∣3x1x3F2∣3x2x3f1∣3x1x3f2∣3x2x3(5.19)

Thus,

fx1x2x3=fx1x2x3fx2x3=c12∣3F1∣3x1x3F2∣3x2x3fx1x3fx2x3fx2x3=c12∣3F1∣3F2∣3f1∣3(5.20)

Alternatively, f(x₁| x₂, x₃)fx1x2x3 may be also written as follows:

Equations (5.20) and (5.21) can be further decomposed as follows:

From the expression of the appropriate pair-copula, a conditional marginal density function can be expressed in a general form as follows:

where v is a d-dimensional vector; v_jvj is one arbitrarily chosen component of v; and v_−jv−j denotes the v vector except v_jvj, i.e., v_−j = v\v_jv−j=v\v_j.

Under appropriate conditions, a multivariate probability density function may be expressed through the product of pair-copulas, acting on several different conditional probability distributions (Aas et al., 2009).

Joe (1997) showed a conditional marginal distribution for the appropriate pair-copula for every j as

Fxv=∂Cx,vj∣v−jFxv−jFvjv−j∂Fvjv−j(5.24)

where C_{x, vj ∣ v−j}Cx,vj∣v−j is a bivariate copula function with the conditional marginals. For the special case where v is univariate, Equation (5.24) can be rewritten as follows:

Fxv=∂Cx,vFXxFVv∂FVv(5.25)

In Equation (5.25), when xx and vv are copula random variables (i.e., the margins following the uniform [0,1] as f(x) = f(v) = 1, F_X(x) = x, F_V(v) = vfx=fv=1,FXx=x,FVv=v), Equation (5.25) can be rewritten as follows:

hxvΘ=Fxv=∂Cx,vxvΘ∂v(5.26)

where the second variable of h(⋅)h⋅ function represents the conditional variable, and ΘΘ denotes the set of copula parameters to model the joint distribution function of xx and vv. Letting u = xu=x, Equation (5.26) is essentially the conditional copula function of C(u| V = v; Θ)CuV=vΘ.

Example 5.13 Derive the hh function for the bivariate Gumbel–Hougaard copula.

Solution: As seen in the previous chapters, the bivariate Gumbel–Hougaard copula can be written as follows:

Cu1u2θ=e−(−lnu1θ+−lnu2)θ1θ

Then the hh function, i.e., h(u₁, u₂, θ)hu1u2θ, can be expressed as follows:

hu1u2θ=Fu1U2=u2θ=∂Cu1u2θ∂u2=Cu1u2u2−lnu2θ−1−lnu1θ+−lnu2θ^ˆ1θ−1=e−−lnu1θ+−lnu2θ1θ−lnu2θ−1−lnu1θ+−lnu2θ1θ−1u2

5.3.2 Vines

High-dimensional distributions have a significant number of possible pair-copula constructions. The regular vine, introduced by Bedford and Cooke (2001, 2002), is used to organize the general structure and embrace a large number of possible pair-copula decompositions. Two special types of regular vines, the C-vine and the D-vine (Kurowicka and Cooke, 2004), are given in the form of a nested set of trees and are used to decompose the multivariate density function. Figure 5.8 shows one sample specification corresponding to a five-dimensional D-vine that can be explained with Table 5.3.

Figure 5.8 A D-vine with five variables, four trees, and 10 edges.