7 – Non-Archimedean Copulas




Abstract




Meta-elliptical copulas are derived from elliptical distributions. Kotz and Nadarajah (2001) and Nadarajah (2006) made solutions of meta-elliptical copulas available. In this chapter, we will review the definition and probability distributions as well as other properties of meta-elliptical copulas.





7 Non-Archimedean Copulas Meta-Elliptical Copulas




7.1 Meta-Elliptical Copulas



7.1.1 d-Dimensional Symmetric Elliptical Type Distribution


In previous chapters, we have discussed symmetric and asymmetric (i.e., nested) Archimedean copulas for multivariate modeling (i.e., d ≥ 3d≥3). We have shown that (i) symmetric multivariate Archimedean copulas require that all the correlated variables share the same dependence structure, and (ii) the nested asymmetric multivariate copulas still require that some variables share the same dependence structure. Compared to the symmetric and nested asymmetric Archimedean copulas, the meta-elliptical copulas are more flexible than the symmetric (or nested) Archimedean copulas for modeling multivariate hydrological variables.


Following Genest et al. (2007), a d-dimensional random vector z (z = [z1, …, zd]Tz=z1…zdT) is said to have an elliptical joint distribution, i.e., ℰd(μ, Σ, g)ℰdμΣg with mean vector μ(d × 1)μd×1, covariance matrix Σ(d × d)Σd×d, and generator g : [0, ∞)→[0, ∞)g:0∞→0∞, if there exists a stochastic representation, as follows:



z = μ + rAu
z=μ+rAu(7.1)

where r ≥ 0r≥0 is a random variable with the probability density function as


fgr=2πd2Γd2rd−1gr2(7.1a)

u (independent of r) is uniformly distributed on the sphere as follows:


Sd=u1…ud∈ℝd:u12+u22+…+ud2=1(7.1b)

A is Cholesky decomposition of Σ, i.e., AAT = ΣAAT=Σ


and the joint probability density function of z can be written as follows:


Σ−12gz−μTΣ−1z−μ(7.2)

In Equations (7.1a) and (7.2), g(⋅)g⋅ is a scale function uniquely determined by the distribution of r and referred to as the probability density function generator. Common d-dimensional symmetric elliptical type distribution generators are given in Table 7.1.




Table 7.1. Common probability density function generators [g(t)gt] for elliptical copulas.




























Copula g(t)gt
Normal 2π−d2exp−t2
Student πv−d2Γd+v2Γd21+tv−d+v2
Cauchy π−d2Γd+12Γ121+t−d+12
Kotza sΓd2r2N+d−22stN−1exp−rtsπd2Γ2N+d−22s;r,s>0,2N+d>2
Pearson type II Γd2+m+1πd2Γm+11−tm;t∈−11,m>−1
Pearson type VIIb ΓNΓN−d2πmd21+tm−N;N>d2,m>0


Notes:




a Kotz type copula reduces to normal copula if N = s = 1, r = 1/2N=s=1,r=1/2.



b Pearson type VII copula reduces to Cauchy copula if m = 1, N = 3/2m=1,N=3/2 and reduces to Student copula if N=m2+1.


To build the meta-elliptical copula using the g(⋅)g⋅ function (listed in Table 7.1) and Equation (7.2), we should note that there is one limitation of these elliptical distributions, that is, the scaled variables z1σ11,z2σ22,…,zdσdd are identically distributed with the density function as follows:


qgzkσkk=x=πΓd−12∫u2∞y−x2d−12−1gydy;k=1,…,d(7.3)

and the CDF of the scaled variables given as follows:


Qgzkσkk≤x=12+πd−12Γd−12∫0x∫u2∞y−u2d−12−1gydydu(7.4)

From Equations (7.3) and (7.4), it is known that qg(x) = qg(−x)qgx=qg−x and Qg(x) = 1 − Qg(x) forx>0Qgx=1−Qgxforx>0.




Example 7.1 Derive the d-dimensional multivariate normal density function: z = [z1, …, zd]z=z1…zd.


Solution: As listed in Table 7.1, the probability density function generator for the multivariate normal distribution is gt=2π−d2exp−t2. Applying Equation (7.2), we have the following:


fz=Σ−122π−d2exp−z−μTΣ−1z−μ2,z~ℰdμΣg(7.5)

If μ = 0μ=0 in Equation (7.5), we have


fz=Σ−122π−d2exp−zTΣ−1z2,z~ℰd0Σg(7.6a)

By applying Equation (7.1), we have zTΣ−1z = r2(Au)TΣ−1(Au) = r2zTΣ−1z=r2AuTΣ−1Au=r2 and Equation (7.6a) may be rewritten as follows:


fz=Σ−122π−d2exp−r22,z~ℰd0Σg(7.6b)

where


Σ=ρ11⋯ρ1d⋮⋱⋮ρd1⋯ρdd,ρii=1;∣ρij∣<1,i≠j;i,j=1,..,d, correlation matrix.




Example 7.2 Derive the d-dimensional multivariate Cauchy density function for z = [z1, …, zd]z=z1…zd.


Solution: Using the probability density function generator for the multivariate Cauchy distribution listed in Table 7.1:


gt=π−d2Γd+12Γ121+t−d+12

Applying Equation (7.2), we have the following:


fz=Σ−122−d2π−dΓd+12Γ121+z−μTΣ−1z−μ2−d+12,z~ℰdμΣg(7.7)

Similarly, if μ = 0μ=0, we have from Equation (7.7) the following:


fz=Σ−122−d2π−dΓd+12Γ121+zTΣ−1z2−d+12,z~ℰd0Σg(7.7a)

Or equivalently


fz=Σ−122−d2π−dΓd+12Γ121+r22−d+12,z~ℰd0Σg(7.7b)

Without loss of generality, we will only investigate the case ℰd(0, Σ, g)ℰd0Σg. Let z = [z1, z2, …, zd]Tz=z1z2…zdT be a random vector with each component zizi with given continuous PDF fi(zi)fizi and CDF Fi(zi)Fizi. Suppose


xi=Qg−1Fizi,i=1,2,…d(7.8)

where Qg−1 is the inverse of QgQg.


Then, the probability density function of z is given by



f(z1, …, zd) = f(x1, …, xd) ∣ J∣
fz1…zd=fx1…xd∣J∣(7.9)

where the Jacobian matrix J is given as follows:


J=∂x1∂z1⋯∂xd∂zd⋮⋱⋮∂x1∂zd⋯∂xd∂zd

Since xi=Qg−1Fizi, we have ∂xi∂zj=dxidzi,i=j0,i≠j. Rewriting matrix J, we have the following:


J=dx1dz1⋯0⋮⋱⋮0⋯dxddzd;J=dx1dz1⋅dx2dz2⋯dxddzd=∏i=1ddxidzi

From xi=Qg−1Fizi, we have Fi(zi) = Qg(xi)Fizi=Qgxi. Differentiation on both sides leads to fi(zi)dzi = qg(xi)dxifizidzi=qgxidxi; dxidzi=fiziqgxi=fiziqg(Qg−1Fizi⇒J=∏i=1dfiziqgQg−1Fizi


Then, we have the following:


fz1z2…zd=fx1x2…xd∏i=1dfiziqg[Qg−1(Fizi]=fx1…xd∏i=1dqgxi∏i=1dfizi(7.10)

For x = (x1, …, xd)T~ℰd(0, Σ, g)x=x1…xdT~ℰd0Σg, we have the following:


fx1…xd=Σ−12gxTΣ−1x(7.11)

Inserting Equation (7.11) in Equation (7.10), we have the following:


fz1…zd=Σ−12gxTΣ−1x∏i=1dqgxi∏i=1dfizi(7.12)

Using ℋℋ to represent the d-variant probability density function as


ℋQg−1F1z1…Qg−1Fdzd=Σ−12gxTΣ−1x∏i=1dqgxi

Equation (7.12) may be written as follows:


fz1…zd=ℋQg−1F1z1…Qg−1Fdzd∏i=1dfizi(7.12a)

To this end, the d-dimensional random vector z is said to have a meta-elliptical distribution, if its probability density function is given by Equation (7.12). Denote x~ℳℰd(0, Σ, g; F1, …, Fd)x~ℳℰd0ΣgF1…Fd. The function ℋQg−1F1z1…Qg−1Fdzd is referred to as the probability density function weighting function. The class of meta-elliptical distributions includes various distributions, such as elliptically contoured distributions, the meta-Gaussian distributions, and various asymmetric distributions. The marginal distributions Fi. can be arbitrarily chosen (Fang et al., 2002). The meta-elliptical distributions allow for the possibility of capturing tail dependence (Joe, 1997), which will be discussed later.



7.1.2 Bivariate Symmetric Elliptical Type Distribution


Suppose x~ℳℰ2(0, Σ, g)x~ℳℰ20Σg, we have the following:


Σ=1ρρ1,Σ−1=11−ρ2−ρ1−ρ2−ρ1−ρ211−ρ2


x1x2Σ−1x1x2=x12+x22−2ρx1x21−ρ2

Vector x has the following probability density function:


fx1x2Σg=11−ρ2gx12+x22−2ρx1x21−ρ2(7.13)

The marginal PDF and CDF of x are


qgx=∫x2∞y−x2−12gydy(7.14)

Qgx=12+∫x2∞arcsinxydy(7.15)

A two-dimensional random vector (x1, x2)x1x2 follows an elliptically contoured distribution, if its joint PDF takes on the form of Equation (7.13). Its copula function can be written as follows:


CXuv=Fx1x2=11−ρ2∫−∞Qg−1u∫−∞Qg−1vgs2+t2−2ρst1−ρ2dsdt(7.16)

where u=F1x1,v=F2x2,s=Qg−1u,t=Qg−1v.


The copula density function can be given as follows:



cX(u, v) = ℋ(s, t; ρ)
cXuv=ℋstρ(7.17)

where


ℋstρ=fstρqgsqgt(7.18)

Now, let z = (z1, z2)T~ℳℰ2(0, Σ, g; F1, F2)z=z1z2T~ℳℰ20ΣgF1F2. Its probability density function may then be expressed as follows:


fx1x2=ℋQg−1F1x1Qg−1F2x2f1x1f2x2(7.19)

Take simple examples to illustrate the preceding.



Symmetric Kotz Type Distribution

Let x be distributed according to a bivariate symmetric Kotz type distribution. Inserting the density generator for Kotz type distribution (listed in Table 7.1) in Equation (7.2), we obtain the joint probability density function as follows:


fx1x2=srNsx12+x22−2ρx1x2N−1πΓNs1−ρ2N−12exp−rx12+x22−2ρx1x21−ρ2s(7.20)

where r>0, s>0, N>0r>0,s>0,N>0 are the parameters.


The marginal PDF (i.e., q1(x)q1x) can be written as follows:


q1x=2srNsπΓNs∫0∞t2+x2N−1exp−rt2+x2sdt(7.21)

The corresponding CDF (i.e., Q1(x)Q1x) can be written as follows:


Q1x=12+2srNsπΓNs∫0x∫0∞t2+x2N−1exp−rt2+x2sdtdx(7.22)

Then, the copula probability density function can be given as follows:


cuv=fQ1−1uQ1−1vq1Q1−1uq2Q2−1v(7.23)



Example 7.3 Show that the bivariate Kotz type distribution converges to the bivariate Gaussian distribution as noted in Table 7.1, i.e., N = s = 1, r = 1/2N=s=1,r=1/2.


Solution: Substituting N=s=1,r=12 into the probability density function generator of symmetric Kotz type distribution, we have


g2t=sΓd2r2N+d−22stN−1exp−rtsπd2Γ2N+d−22s=texp−t/22π,d=2

Comparing with the probability density function generator for the normal copula in the bivariate case, we have the following:


g2N=2π−1exp−t2

Now we show that the bivariate Kotz type distribution reduces to the bivariate normal distribution if N=s=1,andr=12. The same conclusion is reached for higher dimensional cases.




Example 7.4 Compute the copula density function for symmetric Kotz type distribution with the information given as N = 2.0, s = 1.0, r = 0.5, ρ = 0.1, u = 0.4, v = 0.3N=2.0,s=1.0,r=0.5,ρ=0.1,u=0.4,v=0.3.


Solution: Using Equation (7.22), we can calculate Q−1(u), Q−1(v)Q−1u,Q−1v numerically as follows:



Q−1(0.4) =  − 0.4843; Q−1(0.3) =  − 0.9158
Q−10.4=−0.4843;Q−10.3=−0.9158

Using Equation (7.20), we can compute the joint density function as follows:



f(Q−1(0.4), Q−1(0.3)) = f(−0.4843, −0.9158) = 0.0484
fQ−10.4Q−10.3=f−0.4843−0.9158=0.0484

Using Equation (7.21), we can compute the univariate density as follows:



q(Q−1(0.4)) = q(−0.4843) = 0.2190; q(Q(v)) = q(−0.9158) = 0.2411
qQ−10.4=q−0.4843=0.2190;qQv=q−0.9158=0.2411

Finally, substituting the computed quantities above into Equation (7.23), we have the following:


cuv=c0.40.3=0.04840.2190∗0.2411=0.9160

To further illustrate the shape of the bivariate symmetric Kotz type density function, Figure 7.1 graphs the bivariate density function for the following:




  1. 1. N = 2.0, s = 1.0, r = 0.5, ρ = 0.1N=2.0,s=1.0,r=0.5,ρ=0.1.



  2. 2. N = s = 1, r = 0.5, ρ = 0.5N=s=1,r=0.5,ρ=0.5: bivariate normal distribution.





Figure 7.1 Copula density plots for Kotz type bivariate distribution.



Symmetric Bivariate Pearson Type VII Distribution

The PDF of symmetric bivariate Pearson type VII distribution can be given as follows:


fx1x2=N−1πm1−ρ21+1m1−ρ2×12+x22−2ρx1x2−N(7.24)

where N>1N>1, and m>0m>0 are parameters.


When N=m2+1, Equation (7.24) is the bivariate t-distribution with m degrees of freedom. When m=1,n=32, Equation (7.24) is the bivariate Cauchy distribution.


The marginal PDF of symmetric bivariate Pearson type VII distribution is as follows:


qx=Γn−12πmΓN−11+x2m−N−12(7.25)

The corresponding CDF of symmetric bivariate Pearson type VII distribution can be written as follows:


Qx=Γn−12πmΓN−1∫−∞x1+t2m−N−12dt(7.26)

where x ∈ (−∞, ∞), m>0, N>1.x∈−∞∞,m>0,N>1.


Then, the copula density function c(u, v)cuv can be given as follows:


cuv=ΓN−1ΓNΓN−1221+Q1−1um1+Q1−1vmN−121−ρ2[1+1m1−ρ2Q1−1u2+Q1−1v2−2ρQ1−1uQ1−1vN(7.27)



Example 7.5 Show the following bivariate Pearson type VII distribution cases are true.


Show the following cases are true:




  1. 1. N=m2+1, the bivariate Pearson type VII distribution is the bivariate Student t-distribution with m degrees of freedom.



  2. 2. m=1,N=32, the bivariate Pearson type VII distribution is the bivariate Cauchy distribution.


Solution:




  1. 1. N=m2+1


    When N=m2+1, the probability density function generator of the Pearson type VII distribution may be rewritten as follows:


    g2PVII=Γm2+1Γm2−1πm1+tm−m2+1;N>1,m>0,N=m2+1


    Comparing with the probability density function generator for the bivariate Student t-distribution g2t=πv−1Γv2+1Γv21+tv−v2+1, we show that when N=m2+1, the bivariate Pearson type VII distribution reduces to the bivariate Student t-distribution. The same conclusion is reached for higher-dimensional cases.



  2. 2. m=1,N=32


    When m=1,N=32, the probability density function generator of the Pearson type VII distribution may be rewritten as follows:


    g2PVIIt=Γ32Γ12π1+t−32,m=1,N=32


    Comparing with the probability density function generation for the Cauchy distribution g2Cauchy=π−1Γ32Γ121+t−32, we show that when m=1,N=32, the bivariate Pearson type VII distribution reduces to the bivariate Cauchy distribution. The same conclusion is reached for higher-dimensional cases.




Example 7.6 Compute the Pearson type VII copula density with the information given as follows: m = 0.5,  N = 2.0,  ρ = 0.1, u = 0.4, v = 0.3m=0.5,N=2.0,ρ=0.1,u=0.4,v=0.3.


Solution: Applying Equation (7.26), we can compute Q−1(u), Q−1(v)Q−1u,Q−1v numerically as follows:



Q−1(u) =  − 0.1443; Q−1(v) =  − 0.3086
Q−1u=−0.1443;Q−1v=−0.3086

Substituting

Q−1(u) =  − 0.1443; Q−1(v) =  − 0.3086Q−1u=−0.1443;Q−1v=−0.3086

into Equation (7.24), we can compute the copula density function c(0.4, 0.3)c0.40.3 as c(0.4, 0.3) = 1.1941c0.40.3=1.1941.


To illustrate the shape of Pearson type VII distribution, we graph the Pearson type VII copula density function for the following parameters in Figure 7.2:




  1. 1. m = 0.5,  N = 2.0,  ρ = 0.1m=0.5,N=2.0,ρ=0.1.



  2. 2. m = 3,  N = 2.5,  ρ = 0.2m=3,N=2.5,ρ=0.2: bivariate Student t-distribution with degrees of freedom as 3.



  3. 3. m=1,N=32,ρ=0.2: bivariate Cauchy distribution.





Figure 7.2 Pearson type VII copula density plots.



Symmetric Bivariate Pearson Type II Distribution

The PDF of symmetric bivariate Pearson type II distribution can be expressed as


fx1x2=m+1π1−ρ21−x12+x22−2ρx1x21−ρ2m,∀x1x2Σ−1x1x2T≤10otherewise(7.28)

where m> − 1m>−1.


The marginal PDF can be given as follows:


qx=Γm+2πΓm+321−x2m+12;x∈−11(7.29)

The corresponding CDF can be given as follows:


Qx=Γm+2πΓm+32∫−1×1−t2m+12dt;x∈−11(7.30)

The copula probability density function can then be given as follows:


cuv=m+1Γ2m+32Γ2m+21−11−ρ2Q−1u2+Q−1v2−2ρQ−1uQ−1vm1−ρ21−Q−1u2m+121−Q−1v2m+12(7.31)



Example 7.7 Compute the bivariate Pearson type II copula density function with information given as follows: m =  − 0.5, ρ = 0.1, u = 0.4, v = 0.3m=−0.5,ρ=0.1,u=0.4,v=0.3.


Solution: Applying Equation (7.30), we can compute Q−1(0.4), Q−1(0.3)Q−10.4,Q−10.3 numerically as follows:



Q−1(0.4) =  − 0.2; Q−1(0.3) =  − 0.4
Q−10.4=−0.2;Q−10.3=−0.4

Applying Equation (7.31), we can compute the bivariate Pearson type II copula density function as follows:



c(0.4, 0.3) = 0.7091
c0.40.3=0.7091.

To further illustrate the shape of the bivariate Pearson type II copula density function, we graph the Pearson type II copula density function for the following parameters in Figure 7.3:




  • m =  − 0.5, ρ = 0.1m=−0.5,ρ=0.1; (2) m = 0.5, ρ = 0.2m=0.5,ρ=0.2.





Figure 7.3 Pearson type II copula density function plots.



7.2 Two Most Commonly Applied Meta-Elliptical Copulas


In Section 7.1, we have stated that (1) the symmetric meta-Kotz type distribution reduces to the meta-Gaussian distribution if N = s = 1, r = 0.5N=s=1,r=0.5; (2) the symmetric meta-Pearson distribution reduces to the meta-Student t-distribution if N=m2+1. In this section, we will start to focus on the discussion of two most commonly applied meta-elliptical copulas, and these are meta-Gaussian and meta-Student t copulas.



7.2.1 Meta-Gaussian Copula


A d-dimensional meta-Gaussian copula can be expressed as follows:


Cu1…udΣ=ΦΣΦ−1u1…Φ−1ud=∫−∞Φ−1u1…∫−∞Φ−1ud12πd2Σ12exp−12wTΣ−1wdw(7.32)

where Φ−1(⋅)Φ−1⋅ represents the inverse function of standard normal distribution; ΦΣ(Φ−1(u1), …., Φ−1(ud))ΦΣΦ−1u1….Φ−1ud represents multivariate standard normal distribution function; ΣΣ represents the correlation matrix; Σ=1⋯ρ1d⋮⋱⋮ρd1⋯1,


ρij=1i=jρjii≠j,ρij=sinπτij2,τi,j the rank correlation coefficient;


d the dimension of continuous multivariate random variables; and w the integral matrix: w = [w1, …, wd]Tw=w1…wdT.


Let gw1…wd=12πd2Σ12exp−12wTΣ−1w,x1=Φ−1u1,…,xd=Φ−1ud, Equation (7.32) may be rewritten as follows:


Cu1…ud=∫−∞x1⋯∫−∞xdgw1…wddw1…dwd(7.32a)

and its copula density function can be given as follows:


cu1…udΣ=∂d∂u1…∂udCu1…udΣ=∂d∂u1…∂ud∫−∞Φ−1u1⋯∫−∞Φ−1ud12πd2Σ12exp−12wTΣ−1wdw(7.33)

or equivalently


cu1…udΣ=∂d∂u1…∂ud∫−∞x1⋯∫−∞xdgw1…wddw1…dwd(7.33a)

Applying the partial derivative rule of inverse function,


u1=Φx1…ud=Φxd⇒du1dx1=ϕx1…duddxd=ϕxd⇒∂x1∂u1=dx1du1=1du1dx1=1ϕx1…∂xd∂ud=dxddud=1duddxd=1ϕxd(7.34)

In Equation (7.34), Φ(⋅)Φ⋅ is the CDF of standard normal distribution: Φx=∫−∞x12πe−t22dt; and ϕ(⋅)ϕ⋅ is the PDF of standard normal distribution: ϕx=12πe−x22.


Now substituting Equation (7.34) back into Equation (7.32) or (7.32a), we can calculate the partial derivatives for the d-dimensional meta-Gaussian copula in what follows.



First-Order Partial Derivative

Using ∂C∂u1 as an example, the first-order partial derivative of the meta-Gaussian copula may be derived as follows:


∂C∂u1=∂∂u1∫−∞x1…∫−∞xdgw1…wddw1…dwd=∂∂x1∫−∞x1…∫−∞xdgw1…wddw1…dwd∂x1∂u1=1ϕx1∫−∞x2…∫−∞xdgx1…wddw2…dwd(7.35)


Second-Order Partial Derivative

Using ∂2C∂u1∂u2 as an example, the second-order partial derivative of the meta-Gaussian copula may be derived as follows:


∂2C∂u1∂u2=∂∂u2∂C∂u1=∂∂u21ϕx1∫−∞x2…∫−∞xdgx1…wddw2…dwd=∂∂x21ϕx1∫−∞x2…∫−∞xdgx1…wddw2…dwd∂x2∂u2=1ϕx1ϕx2∫−∞x3…∫−∞xdgx1x2…wddw3…dwd(7.36)


dth-Order Partial Derivative

Repeating the derivative d-times, we obtain the meta-Gaussian copula density function as follows:


cu1…udΣ=∂du1…udΣ∂u1…∂ud=1ϕx1…ϕxdgx1…xd(7.37)

Let ς = [x1, …, xd]T = [Φ−1(u1), …, Φ−1(ud)]Tς=x1…xdT=Φ−1u1…Φ−1udT. Equation (7.37) may be rewritten as follows:


cu1…udΣ=112πe−x122…12πe−xd22·12πd2Σ12exp−12ςTΣ−1ς

=112πe−Φ−1u122…12πe−Φ−1ud22·12πd2Σ12exp−12ςTΣ−1ς

=1∏i=1d12πe−Φ−1ui22·12πd2Σ12exp−12ςTΣ−1ς

=Σ−12·1∏i=1de−Φ−1ui22exp−12ςTΣ−1ς(7.38)

Note that in Equation (7.38), ∏i=1de−Φ−1ui22 may be rewritten as follows:


∏i=1de−Φ−1ui22=exp−Φ−1u12+…+Φ−1ud2(7.38a)

Φ−1u12+…+Φ−1ud2=Φ−1u1…Φ−1udΦ−1u1…Φ−1ud=ςTς(7.38b)

Substituting Equations (7.38a) and (7.38b) into Equation (7.38), Equation (7.38) may be simplified as follows:


cu1…udΣ=Σ−121e−ςTς2exp−12ςTΣ−1ς=Σ−12exp−12ςTΣ−1ς+ςTς2(7.39)

Recall that ςTς = ςTIςςTς=ςTIς, where I is d by d identity matrix. Equation (7.39) may also be rewritten as follows:


cu1…udΣ=Σ−12exp−12ςTΣ−1−Iς(7.39a)



Example 7.8 (Bivariate meta-Gaussian copula): Compute the bivariate meta-Gaussian copula and its copula density function with the given information: Σ=10.20.21,u1=0.4,u2=0.3, and show the first-order derivative of the bivariate meta-Gaussian copula.


Solution: Applying Equation (7.32) for d = 2d=2, we have the bivariate meta-Gaussian copula as follows:


Cu1u2Σ=ΦΣΦ−1u1Φ−1u2=∫−∞Φ−1u1∫−∞Φ−1u212πΣ12exp−12x1x2Σ−1x1x2Tdx1dx2(7.40)

From standard normal distribution, we have the following:



Φ−1(0.4) =  − 0.2533; Φ−1(0.3) =  − 0.5244
Φ−10.4=−0.2533;Φ−10.3=−0.5244.

Σ−1=1.0417−0.2083−0.20831.0417;Σ=0.96.

Substituting Φ−1(0.4), Φ−1(0.3), Σ−1Φ−10.4,Φ−10.3,Σ−1 and ∣Σ∣∣Σ∣ into Equation (7.40), we have the following:



C(0.4, 0.3; Σ) = 0.1474
C0.40.3Σ=0.1474.

Applying Equation (7.39a) for d = 2d=2, we have the meta-Gaussian copula density function as follows:


cu1u2Σ=Σ−12exp−12Φ−1u1Φ−1u2Σ−1−IΦ−1u1Φ−1u2;I=1001=Σ−12exp−12x1x2Σ−1−Ix1x2(7.40a)

Substituting Φ−1(0.4), Φ−1(0.3), Σ−1Φ−10.4,Φ−10.3,Σ−1 and ∣Σ∣∣Σ∣ into Equation (7.40a), we have the following:



c(0.4, 0.3; Σ) = 1.0419
c0.40.3Σ=1.0419.

Applying Equation (7.35) for d = 2d=2, we have the first-order derivative of the bivariate meta-Gaussian copula function as follows:


∂C∂u1=1ϕx1∫−∞x2gx1w2dw2=1ϕx1∫−∞x212πΣ12exp−12x1w2Σ−1x1w2dw2(7.41)

Substituting Σ=1−ρ2,Σ−1=1−ρ−ρ11−ρ2 into Equation (7.41), we have the following:


∂C∂u1=1ϕx1∫−∞x212π1−ρ2exp−121−ρ2×12−2ρx1w2+w22dw2=1ϕx112π1−ρ2exp−x1221−ρ212π∫−∞x2exp−121−ρ2w22−2ρx1w2dw2(7.41a)

In Equation (7.41a),

12π∫−∞x2exp−121−ρ2w22−2ρx1w2dw2

may be further simplified as follows:


12π∫−∞x2exp−121−ρ2w22−2ρx1w2dw2

=12π∫−∞x2exp−121−ρ2w2−ρx12−ρ2x12dw2

=12πexpρ2×1221−ρ2∫−∞x2exp−12w2−ρx11−ρ22dw2

Let y=w2−ρx11−ρ2. We have the following:


12π∫−∞x2exp−12w2−ρx11−ρ22dw2=1−ρ22π∫−∞x2−ρx11−ρ2exp−y22dy=1−ρ2Φx2−ρx11−ρ2

Finally, Equation (7.41a) is rewritten as follows:


∂C∂u1=1ϕx112π1−ρ2exp−x1221−ρ2expρ2×1221−ρ21−ρ2Φx2−ρx11−ρ2=Φx2−ρx11−ρ2(7.42)

or equivalently


∂C∂u1=ΦΦ−1u2−ρΦ−1u11−ρ2(7.42a)

To further illustrate the shape of meta-Gaussian copula and its density function, Figure 7.4 graphs the meta-Gaussian copula and its density function with the use of parameters given in this example.





Figure 7.4 Meta-Gaussian copula and its copula density plots.




Example 7.9 (Trivariate meta-Gaussian copula): compute the trivariate meta-Gaussian copula and its density function.


Compute the copula and its density function with information given as follows:


Σ=10.20.60.210.40.60.41,u1=0.4;u2=0.3;u3=0.8.

Also, show the first- and second-order derivatives of the trivariate meta-Gaussian copula.


Applying Equation (7.32) for d = 3d=3, we have the following:



C(u1, u2, u3; Σ)
Cu1u2u3Σ

=∫−∞Φ−1u1∫−∞Φ−1u2∫−∞Φ−1ud12π32Σ12exp−12x1x2x3Σ−1x1x2x3dx1dx2dx3(7.43)

From standard normal distribution, we have the following:



Φ−1(0.4) =  − 0.2533, Φ−1(0.3) =  − 0.5244, Φ−1(0.7) = 0.8416
Φ−10.4=−0.2533,Φ−10.3=−0.5244,Φ−10.7=0.8416

∣Σ∣∣Σ∣ and Σ−1Σ−1 are calculated as follows:


Σ=0.5360;Σ−1=1.56720.0746−0.97010.07461.1940−0.5224−0.9701−0.52241.7910

Integrating Equation (7.43) with the calculated quantity numerically, we have the following:



C(0.4, 0.3, 0.8; Σ) = 0.1450; c(0.4, 0.3, 0.8; Σ) = 0.6309
C0.40.30.8Σ=0.1450;c0.40.30.8Σ=0.6309.

Applying Equations (7.35) for d = 3d=3, we have the first-order derivative of trivariate meta-Gaussian copula as follows:


∂Cu1u2u3∂u1=1ϕx1∫−∞x2∫−∞x3gx1w2w3dw2dw3=1ϕx1∫−∞x2∫−∞x312π32Σ12exp−12x1w2w3Σ−1x1w2w3dw2dw3(7.44)

Let Σ=1ρ12ρ13ρ121ρ23ρ13ρ231, we have the following:


Σ−1=1∣Σ∣1−ρ232ρ13ρ23−ρ12ρ12ρ23−ρ13ρ13ρ23−ρ121−ρ132ρ12ρ13−ρ23ρ12ρ23−ρ13ρ12ρ13−ρ231−ρ122

where Σ=1−ρ122−ρ132−ρ232+2ρ12ρ13ρ23


The conditional copula defined in Equation (7.44) follows the bivariate normal distribution that is derived in what follows. Under the condition, i.e., U1 = u1U1=u1 or equivalently X1 = x1X1=x1, we first partition the random variable, ΣΣ, and Σ−1Σ−1 as follows:


x1w2w3=x1w,wherew=w2w3(7.44a)

Σ=1ρ12ρ13ρ121ρ23ρ13ρ231=Σ11Σ12Σ21Σ22(7.44b)

where Σ11=1,Σ12=Σ21T=ρ12ρ13,Σ22=1ρ23ρ231


Σ−1=V11V12V21V22(7.44c)

where V11=1Σ1−ρ232,V12=V21T=1∣Σ∣ρ13ρ23−ρ12ρ12ρ23−ρ13


V22=1∣Σ∣1−ρ132ρ12ρ13−ρ23ρ12ρ13−ρ231−ρ122

Substituting Equations (7.44a), (7.44b), and (7.44c) into Equation (7.44), we have the following:


x1x2x3Σ−1x1x2x3=x1wTV11V12V21V22x1w=x12V11+x1V12w+wTV21x1+wTV22w(7.44d)

After some algebra, Equation (7.44d) may be rewritten as follows:


x1x2x3Σ−1x1x2x3=w−aTV22w−a+b(7.44e)

where a=−V22−1V21x1,b=x12V11−V21TV22−1V21


Equation (7.44e) can be rewritten as follows:


x1x2x3Σ−1x1x2x3=w+V22−1V21x1TV22w+V22−1V21x1+x12V11−V21TV22−1V21(7.44f)

Substituting Equation (7.44f) back into Equation (7.44), we have the following:


∂Cu1u2u3∂u1=1ϕx1∫−∞x2∫−∞x312π32Σ12exp−12x1w2w3Σ−1x1w2w3dw2dw3=1ϕx1∫−∞x2∫−∞x3expw+V22−1V21x1TV22w+V22−1V21x1+x12V11−V21TV22−1V212π32Σ12dw2dw3∝exp−12w+V22−1V21x1TV22w+V22−1V21x1~BVN−V22−1V21x1V22−1(7.45)

where V22−1=1Σ21−ρ122ρ23−ρ12ρ13ρ23−ρ12ρ131−ρ132, −V22−1V21x1=1Σρ12×1ρ13×1.


Similarly, we can derive the second-order derivative of the trivariate meta-Gaussian copula. The second-order derivative of the triavariate meta-Gaussian copula follows the univariate normal distribution, i.e., ∂Cu1u2u3∂u1∂u2~N−x1ρ12ρ23−ρ13+x2ρ12ρ13−ρ231−ρ122Σ1−ρ122.



7.2.2 Meta-Student t Copula


A d-dimensional meta-Student t copula can be expressed as follows:


Cu1…udΣν=TΣ,νTν−1u1…Tν−1ud=∫−∞Tν−1u1…∫−∞Tν−1udΓν+d2Γν21πνd2Σ121+wTΣ−1wν−ν+d2dw(7.46)

where




  • Tν−1⋅ represents the inverse of the univariate Student t distribution with νν degrees of freedom.



  • TΣ,νTν−1u1…Tν−1ud represents the multivariate Student t distribution with correlation matrix ΣΣ and νν degrees of freedom in which Σ=1⋯ρ1d⋮⋱⋮ρd1⋯1,ρij=1,i=jρji,i≠j.



  • d represents the dimension of variables; and w represents the integral matrix: w = [w1, …, wd]Tw=w1…wdT.


Let gw=Γv+d2Γν21πνd2Σ121+wTΣ−1wν−ν+d2 and x1=Tν−1u1,…xd=Tν−1ud. Equation (7.46) can then be rewritten as follows:


Cu1…udΣν=∫−∞x1…∫−∞xdgw1…wddw1…dwd(7.47)

Its copula density function can then be written as follows:


cu1…udΣν=∂d∂u1…∂udCu1…udΣν=∂d∂u1…∂ud∫−∞Tν−1u1…∫−∞Tν−1udΓν+d2Γν21πνd2Σ121+wTΣ−1wν−ν+d2dw(7.48)

or equivalently


cu1…udΣν=∂d∂u1…∂ud∫−∞x1…∫−∞xdgw1…wddw1…dwd(7.48a)

Apply the partial derivative rules of the inverse function:


u1=Tνx1…ud=Tνxd⇒du1dx1=tvx1…duddxd=tvxd⇒∂x1∂u1=dx1du1=1du1dx1=1tνx1…∂xd∂ud=dxddud=1duddxd=1tνxd(7.49)

Now, substituting Equation (7.49) into Equation (7.48) or (7.48a), we can compute the partial derivatives for the d-dimensional meta-Student t copula. Similar to the d-dimensional meta-Gaussian copula, we will calculate the conditional copula by partitioning the random vector X = [X1, …, Xd]TX=X1…XdT, its correlation matrix ΣΣ, and inverse function Σ−1Σ−1 as follows:




  • Partitioning X, Σ, Σ−1Σ,Σ−1 as follows:


    X=X1X2;Σ=Σ11Σ12Σ21Σ22;Σ−1=V=V11V12V21V22(7.50)


where


X1 = [X1, …, Xd1]TX1=X1…Xd1T (the conditional m-dimensional vector), X2 = [Xd1 + 1, …, Xd]TX2=Xd1+1…XdT; Σ12=Σ21T;V12=V21T;


V11=Σ11−Σ12Σ22−1Σ21−1,d1byd1matrixV12=V21T=−Σ11−1Σ12Σ22−Σ21Σ11−1Σ12−1,d−d1byd1matrix)V22=Σ22−Σ21Σ11−1Σ12−1,d−d1byd−d1matrix)(7.50a)

Then, XTΣ−1XXTΣ−1X in Equation (7.48) can be rewritten as follows:


XTΣ−1X=X1TV11X1+X1TV12X2+X2TV21X1+X2TV22X2

=X1TV11X1+2X1TV12X2+X2TV22X2(7.51)

Expressing the square in X2X2, we can compute the conditional distribution as follows:



XTΣ−1X = (X2 − m)TM(X2 − m) + C
XTΣ−1X=X2−mTMX2−m+C(7.51a)

where


M=V22;C=X1TV11−V21TV22−1V21X1=X1TΣ11−1X1(7.51b)

m=−V22−1V21X1=R21R11−1X1(7.51c)



  • Apply the conditional density function fXX1=fXfX1; after some algebra, we have the following:



    X ∣ X1~T(X2; μ2 ∣ 1, Σ2 ∣ 1, ν2 ∣ 1)
    X∣X1~TX2μ2∣1Σ2∣1ν2∣1    (7.52)


where


TT represents the multivariate (or univariate) Student t distribution;


μ2∣1=m=−V22−1V21X1=R21R11−1X1Σ2∣1=ν+X1TΣ11−1X1ν+d1Σ22−Σ21Σ11−1Σ12ν2∣1=v+d1(7.52a)


First-Order Partial Derivative


∂C∂u1=∂∂u1∫−∞x1…∫−∞xdgw1…wddw1…dwd=∫−∞x2…∫−∞xdgx1w2…wdtνx1dw2…dwd=∫−∞x2…∫−∞xdgx1w2…wddw2…dwdtνx1(7.53)


In Equation (7.53), gx1w2…wdfx1 is the conditional density function given x1x1. Applying Equations (7.50)–(7.52), we have the conditional copula, which follows the d – 1 cumulative multivariate (or univariate if d = 2) Student t distribution with the following parameters:


Σ=1⋯ρ1d⋮⋱⋮ρd1⋯1;Σ11=1,Σ12=Σ21T=ρ12…ρ1d,Σ22=1⋯ρ2d⋮⋱⋮ρd2⋯1(7.54)


μ2 ∣ 1 = (Σ22 − Σ21Σ12)(Σ12(Σ22 − Σ21Σ12)−1)Tx1
μ2∣1=Σ22−Σ21Σ12Σ12Σ22−Σ21Σ12−1Tx1(7.54a)

Σ2∣1=ν+x12ν+1Σ22−Σ21Σ12(7.54b)


ν2 ∣ 1 = ν + 1
ν2∣1=ν+1(7.54c)


Second-Order Partial Derivative


∂2C∂u1∂u2=∂∂u21tνx1∫−∞x2…∫−∞xdgx1w2…wddw2…dwd=1tνx1tνx2∫−∞x3…∫−∞xdgx1x2…wddw3…dwd(7.55)


Similar to the first-order partial derivative for meta-Student t copula, the second-order partial derivative again follows the d-2 cumulative multivariate (or univariate if d = 3) Student t distribution. Based on the derivations given in Equations (7.50)–(7.52), the parameters of the conditional copula are derived in what follows:


Equation (7.50) is rewritten as follows:


X=X1X2;X1=x1x2,X2=x3⋮xd(7.56)

Σ11=1ρ12ρ121,Σ12=Σ21T=ρ13…ρ1d,Σ22=1⋯ρ3d⋮⋱⋮ρd3⋯1(7.56a)

Substituting Equation (7.56) back into Equation (7.52), we obtain the parameters for the conditional Student t distribution as follows:


μ2∣1=Σ22−Σ21Σ11−1Σ12Σ12Σ22−Σ21Σ11−1Σ12−1Tx1x2(7.56c)

Σ2∣1=ν+x1x2Σ11−1x1x2ν+2Σ22−Σ21Σ11−1Σ12(7.56d)

ν2∣1=ν+2(7.56e)


dth-Order Partial Derivative

Using the same approach, the PDF of d-dimensional meta-Student t copula can be obtained as follows:


cu1…udΣν=∂dCu1…udΣν∂u1…∂ud=1tνx1⋯tνxdgx1…xd(7.57)

Let X=x1…xdT=Tν−1u1…Tν−1udT. Then, g(x1, …, xd)gx1…xd can be given as follows:


gX=gx1…xd=Γν+d2Γν21πνd2Σ121+XTΣ−1Xν−ν+d2(7.57a)

cu1…udΣν=Γν+d2∏i=1dtνxiΓν21πνd2Σ121+XTΣ−1Xν−ν+d2(7.57b)

Substituting tνxi=Γν+12Γν2πν121+xi2ν−ν+12 into Equation (7.57b), we have the following:


cu1…ud=Γν+d2Γd−1ν2Γdν+12Σ121+XTΣ−1Xν−v+d2∏i=1d1+xiν−ν+12(7.57c)



Example 7.10 (Bivariate meta-Student t copula): compute the bivariate meta-Student t copula and its density function.


Compute the copula and its density function with the following information:


Σ=10.20.21,ν=2,u1=0.4,u2=0.3.

Also, show the first-order derivative of the bivariate meta-Student t copula.


Solution: For the bivariate meta-Student t copula, let


Σ=1ρρ1,X=x1x2T=Tν−1u1Tν−1u2T

and we have the following:


Σ=1−ρ2;Σ−1=11−ρ21−ρ−ρ1

XTΣ−1X=Tν−1u1Tν−1u21−ρ−ρ1Tν−1u1Tν−1u2T1−ρ2=Tν−1u12−2ρTν−1u1Tν−1u2+Tν−1u221−ρ2

Then, the bivariate meta-Student t copula and its copula density can be expressed as follows:


 C(u1,u2 ; Σ, ν)=TΣ,ν(Tν−1(u1),Tν−1(u2))=∫−∞Tν−1(u1)∫−∞Tν−1(u2)Γ(ν+22)Γ(ν2)1πν|Σ|12(1+wTΣ−1wν)−v+22dw(7.58)

cu1u2Σν=Γν+221+XTΣ−1Xν−ν+22Γν2πνΣ12tνx1tνx2=Γν+22Γν2Γ2ν+121−ρ2121+Tν−1u12−2ρTν−1u1Tν−1u2+Tν−1u22ν1−ρ2−v+221+Tν−1u12ν−ν+121+Tν−1u22ν−ν+12(7.59)

Applying the inverse of univariate Student t distribution with the degrees of freedom (d.f.) = 2, we have the following:


x1=Tν−1u1=T2−10.4=−0.2887;x2=T2−10.3=−0.6172;

The determinant and the inverse of correlation matrix can be computed as follows:


Σ=0.96;Σ−1=1.0417−0.2083−0.20831.0417.

Substituting the computed quantities into Equation (7.58), we have the following:



C(u1, u2; Σ, ν) = TΣ, ν(x1, x2) = 0.1510
Cu1u2Σν=TΣ,νx1x2=0.1510.

Substituting the computed quantities into Equation (7.59), we can compute the copula density function:


c(u1, u2; Σ, ν) = 1.2365cu1u2Σν=1.2365.


Figure 7.5 plots the corresponding copula and its density function.





Figure 7.5 Meta-Student t copula and its density.


In what follows, we give the expression for the first-order derivative of the bivariate meta-Student t distribution.


Applying Equation (7.54a), we have the following:


μ2∣1=1−ρ2ρ1−ρ2−1Tν−1u1=ρTν−1u1(7.60)

Σ2∣1=ν+Tν−1u12ν+11−ρ2(7.60a)


ν2 ∣ 1 = ν + 1
ν2∣1=ν+1(7.60b)

Substituting Equation (7.60) back into Equation (7.52), we have the following:


∂Cu1u2∂u1=Tν+1Tν−1u2−ρTν−1u1ν+Tν−1u12ν+11−ρ2(7.61)

Substituting ν = 2, ρ = 0.2ν=2,ρ=0.2 into Equation (7.61), we have the conditional copula for this example as follows:


∂Cu1u2∂u1=T3T2−1u2−0.2T2−1u10.962+T2−1u123.



Example 7.11 (Trivariate meta-Student t copula): compute the bivariate meta-Student t copula and its density function.


Compute the copula and its density function with the following given information:


Σ=10.20.60.210.40.60.41,ν=2,u1=0.4,u2=0.3,u3=0.8.

Also, show the first- and second-order derivative of the trivariate meta-Student t copula.


Solution: Applying Equation (7.46) for d = 3, we have the following:


Cu1u2u3=∫−∞Tν−1u1∫−∞Tν−1u2∫−∞Tν−1u3Γν+32Γν21πν32Σ121+wTΣ−1wdw(7.62)

From the Student t distribution with d.f. = 2, we have the following:


T2−10.4=−0.2887,T2−10.3=−0.6172,T2−10.8=1.0607.

|Σ|, Σ−1Σ,Σ−1 can be calculated as follows:


Σ=0.5360,Σ−1=1.56720.0746−0.97010.07461.1940−0.5224−0.9701−0.52241.7910

Integrating Equation (7.62) with the computed quantities, we have the following:



C(0.4, 0.3, 0.8; Σ, ν) = 0.1445; c(0.4, 0.3, 0.8; Σ, ν) = 0.4697
C0.40.30.8Σν=0.1445;c0.40.30.8Σν=0.4697.

In the following, we will show the first- and second-order derivatives of the trivariate meta-Student t copula.


First-order derivative of the trivariate meta-Student t copula:


For the trivariate case, Equation (7.54) can be rewritten for X=X1X2,X1=x1;X2=x2x3 as follows:


Σ=1ρ12ρ13ρ121ρ23ρ13ρ231;Σ11=1,Σ12=Σ21T=ρ12ρ13,Σ22=1ρ23ρ231(7.63)

μ2∣1=1−ρ122ρ23−ρ12ρ13ρ23−ρ12ρ131−ρ132ρ12−ρ13ρ23ρ13−ρ12ρ23ΣTx1=1−ρ12ρ12−ρ13ρ23−ρ23−ρ12ρ13ρ13−ρ12ρ23ρ23−ρ12ρ13ρ12−ρ13ρ23−1−ρ132ρ13−ρ12ρ23×1∣Σ∣(7.63a)

Σ2∣1=ν+x12ν+11−ρ122ρ23−ρ12ρ13ρ23−ρ12ρ131−ρ132(7.63b)


ν2 ∣ 1 = ν + 1
ν2∣1=ν+1(7.63c)

Substituting Equations (7.63a)–(7.63c) into Equation (7.52), we have the first-order derivative for the trivariate meta-Student t copula as follows:



C(u2, u3| u1) = BT((X2 − μ2 ∣ 1); Σ2 ∣ 1, ν2 ∣ 1)
Cu2u3u1=BTX2−μ2∣1Σ2∣1ν2∣1(7.63d)

where BT represents the bivariate cumulative Student t distribution.


Furthermore, for this example, we have the following:


μ2∣1=T2−1u10.20.6,Σ2∣1=2+T2−1u130.960.280.280.64,ν2∣1=3.

Cu2u3u1=BTT2−1u2−0.2T2−1u1T2−1u3−0.6T2−1u12+T2−1u130.960.280.280.643.

Second-order derivative of the trivariate meta-student t copula:


In this case, Equation (7.56a) can be rewritten for X=X1X2,X1=x1x2;X2=x3 as follows:


Σ11=1ρ12ρ121,Σ12=Σ21T=ρ13ρ23,Σ22=1(7.64a)

μ2∣1=Σ22−Σ21Σ11−1Σ12Σ12Σ22−Σ21Σ11−1Σ12−1Tx1x2

=(Σ+ρ12ρ13ρ231−ρ122ρ13ρ231−ρ122Σ+ρ12ρ13ρ23x1x2=ρ13×1+ρ23×2(7.64b)

Σ2∣1=ν+x1x2Σ11−1x1x2ν+2Σ22−Σ21Σ11−1Σ12

=ν+x12−2ρ12x1x2+x22ν+21−ρ1222Σ+ρ12ρ13ρ23(7.64c)


ν2 ∣ 1 = ν + 2
ν2∣1=ν+2(7.64d)

Substituting Equations (7.64b)–(7.64d) into Equation (7.52), we have the second-order derivative for the trivariate meta-Student t copula as follows:



C(u3| u1, u2) = T((x3 − μ2 ∣ 1); Σ2 ∣ 1, ν2 ∣ 1)
Cu3u1u2=Tx3−μ2∣1Σ2∣1ν2∣1(7.64e)

Furthermore, for this example, we have the following:


μ2∣1=0.6T2−1u1+0.4T2−1u2

Σ2∣1=0.5842+T2−1u12−0.4T2−1u1T2−1u2+T2−1u223.6864,ν2∣1=4.

Cu3u1u2=T4T2−1u3−0.6T2−1u1+0.4T2−1u20.5842+T2−1u12−0.4T2−1u1T2−1u2+T2−1u223.6864


7.3 Parameter Estimation



7.3.1 Marginal Distributions



Marginal CDF of Symmetric Kotz Type Distribution

From qKotz=2srNsπΓNs∫0∞t2+x2N−1exp−rt2+x2sdt, we can use the Gauss–Laguerre numerical integration method to calculate the marginal CDF of the symmetric Kotz type distribution:


∫0∞fxdx=∫0∞e−xexfxdx≈∑i=1nωxiexifxi≈∑i=1nwxifxi(7.65)

where xixi is the abscissa; ω(xi)ωxi is the weight of abscissas xixi; w(xi)wxi is the total weight of abscissa xixi, w(xi) = ω(xi)exiwxi=ωxiexi; and n is the number of integral nodes. For n = 32, xixi, ω(xi)ωxi and w(xi)wxi are given in Table 7.2.




Table 7.2. Abscissas and weights of Gauss–Laguerre integration.






































































































































































No K Abscissasxixi Weightω(xi)ωxi Total weight w(xi)wxi No K Abscissas

xixi		 Weight

ω(xi)ωxi		 Total weight

w(xi)wxi
1 0.044489 0.109218 0.114187 17 22.63578 4.08E–10 2.764644
2 0.234526 0.210443 0.266065 18 25.62015 2.41E–11 3.228905
3 0.576885 0.235213 0.418793 19 28.87393 8.43E–13 2.920194
4 1.072449 0.195903 0.572533 20 32.33333 3.99E–14 4.392848
5 1.722409 0.129984 0.727649 21 36.1132 8.86E–16 4.279087
6 2.528337 0.070579 0.884537 22 40.13374 1.93E–17 5.204804
7 3.492213 0.031761 1.043619 23 44.52241 2.36E–19 5.114362
8 4.616457 0.011918 1.205349 24 49.20866 1.77E–21 4.155615
9 5.903958 0.003739 1.370222 25 54.35018 1.54E–23 6.198511
10 7.358127 0.000981 1.538776 26 59.87912 5.28E–26 5.347958
11 8.982941 0.000215 1.711646 27 65.98336 1.39E–28 6.283392
12 10.78301 3.92E-05 1.889565 28 72.68427 1.87E–31 6.891983
13 12.76375 5.93E-06 2.073189 29 80.18837 1.18E–34 7.920911
14 14.93091 7.43E-07 2.265901 30 88.73519 2.67E–38 9.204406
15 17.29327 7.63E-08 2.469974 31 98.82955 1.34E–42 11.16374
16 19.85362 6.31E-09 2.642967 32 111.7514 4.51E-48 15.39024

Kotz and Nadarajah (2001) and Nadarajah and Kotz (2005) derived an expression of the hypergeometric function of PDF and CDF of the bivariate symmetric Kotz type distribution and a marginal CDF of the bivariate Pearson type II and VII distributions in the incomplete beta function, respectively.


The PDF and CDF of the bivariate symmetric Kotz type distribution, for z>0z>0, are


qKotzz=r12sexp−rz2sπΓNs∑i=0∞−1iris−12iz2iψ1−Ns+is+12s1−Ns+is+12srz2s(7.66)

where ψψ is the degenerate hypergeometric function given as follows:


ψαβx=Γ1−βΓα−β+1F1αβx+Γβ−1Γαx1−βF11α−β+12−βx(7.66a)

F11αβx=1+∑i=1∞aixibii!;ai=Γa+iΓa,bi=Γb+iΓb(7.66b)

−12i=−12−12−1−12−2⋯−12−i+1i!=−1i22i2ii(7.66c)

The corresponding CDF for z>0z>0


QKotzz=1−1πΓNs∑i=0∞−1i−12iz2i{12i+1ΓNs−r12srisz2i+1ΓNs−is−12s+srNsi2NN2N−2i−1F22Ns−is−12sNsNs−is−12s+1Ns+1−rz2s}(7.67)

where


F22=1+∑i=1∞a1ia2ib1ib2ixii!(7.67a)

Equation (7.67) needs to satisfy Ns−is−12s+1≠0 and Ns+1≠0.


Since Equation (7.67) is an expression of hypergeometric function, it needs to satisfy Ns−is−12s+1≠0, and the numerical solution may experience overflow. Therefore, the Gauss–Laguerre integration and multiple complex Gauss–Legendre integral formulae can be used to compute the marginal PDF and CDF of bivariate symmetric Kotz type distribution, respectively.


For the marginal PDF, the Gauss–Laguerre integration can be used as


qKotzx≈2srNsπΓNs∑k=1qwtketk∑i=1mtk2+xl2N−1exp−rtk2+xl2s(7.68)

where tktk and w(tk)wtk are the abscissa and the weight of the Gauss–Laguerre integration, respectively; m is the integral node; and q is the node of Gauss–Legendre integration. For CDF, we use multiple complex Gauss–Legendre integral formulae (Zhang, 2000):


∫ab∫φxψxfxydydx≈Δx2∑i=1qαiq∑k=1qαkq∑j=0mΔyj2∑l=0njfx˜jiy˜lk(7.69)

where q is the node of Gauss–Legendre integration; a, b are the upper and lower integral limits of variable x; ψ(x) and φ(x)ψxandφx are the upper and lower integral limits of variable y; and m is a positive integer that breaks the interval [a, b] of x into m equal pieces. The width of each piece is Δx=b−am;xj=a+jΔx,j=0,1,…,m;x˜ji=xj+Δx21+x˜iq, x˜iq is the abscissa of ith node of the Gauss–Legendre integration; and njnj is a positive integer that breaks the interval φx˜jiψx˜ji of yy into njnj equal pieces, Δyj=ψx˜ji−φx˜jinj;


yl=φx˜ji+lΔyj,l=0,1,…,nj;y˜lk=yl+1+x˜kqΔyj2; αiqandβq are the abscissas of the ith and kth nodes of the Gauss–Legendre and the Gauss–Laguerre integration, respectively; x˜kq is the abscissa of the kth node of the Gauss–Laguerre integration. From Equation (7.69), we know the integral interval is [0, ∞)0∞. Using the Gauss–Laguerre integration for y, one can get


QKotzx≈12+2srNsπΓNsΔxΔy4∑i=1q∑k=1qαiqβkq∑j=0m∑l=0my˜lk2+x˜ji2N−1exp−ry˜lk2+x˜ji2s(7.70)


Marginal CDF of Symmetric Pearson Type VII Distribution

According to Fang et al. (2002), for z = [x, y]z=xy the bivariate symmetric Pearson type VII distribution can be given as follows:


qPVIIz=qPVIIxy=N−1πm1−ρ21+1m1−ρ2×2+y2−2ρxy−N,N>1,m>0(7.71)

Through integration, the marginal CDF of the symmetric Pearson type VII distribution can be written as follows:


Qp7x=1−ΓN−12πmΓN−1∫x∞1+y2m−N−12dy=ΓN−12πmΓN−1∫−∞x1+y2m−N−12dy(7.72)

On one hand, Equation (7.72) can be solve by applying the Gauss–Laguerre integration to compute the marginal CDF; on the other hand, it can be solved by applying the incomplete beta function (Kotz and Nadarajah, 2001), as follows:


Qp7x=12Imm+x2N−112,x≤01−12Imm+x2N−112,x>0(7.73)

where Ix(a, b)Ixab is the incomplete beta function, as follows:


Ixab=1Bab∫0xta−11−tb−1dt(7.73a)

Bab=∫01ta−11−tb−1dt(7.73b)

Results of the Gauss–Laguerre integration and incomplete beta function results by Kotz and Nadarajah (2001) are very close, as shown in Table 7.3.




Table 7.3. Marginal CDF of the symmetric Pearson type VII distribution (N = 4.0; m = 5.5)




































































































































































































































































































xx qp7(x)qp7x Qp7(x)[1]Qp7x1 Qp7(x)[2]Qp7x2 xx qp7(x)qp7x Qp7(x)[1]Qp7x1 Qp7(x)[2]Qp7x2
−3.0 0.0134 0.0101 0.0101 0.0 0.3998 0.5000 0.5000
−2.9 0.0155 0.0116 0.0116 0.1 0.3972 0.5399 0.5399
−2.8 0.0180 0.0132 0.0132 0.2 0.3897 0.5793 0.5793
−2.7 0.0208 0.0152 0.0152 0.3 0.3777 0.6177 0.6177
−2.6 0.0242 0.0174 0.0174 0.4 0.3616 0.6547 0.6547
−2.5 0.0280 0.0200 0.0200 0.5 0.3422 0.6899 0.6899
−2.4 0.0326 0.0231 0.0231 0.6 0.3202 0.7230 0.7230
−2.3 0.0378 0.0266 0.0266 0.7 0.2965 0.7539 0.7539
−2.2 0.0439 0.0306 0.0306 0.8 0.2719 0.7823 0.7823
−2.1 0.0509 0.0354 0.0354 0.9 0.2471 0.8083 0.8083
−2.0 0.0590 0.0409 0.0409 1.0 0.2228 0.8317 0.8317
−1.9 0.0684 0.0472 0.0472 1.1 0.1993 0.8528 0.8528
−1.8 0.0790 0.0546 0.0546 1.2 0.1771 0.8716 0.8716
−1.7 0.0912 0.0631 0.0631 1.3 0.1565 0.8883 0.8883
−1.6 0.1049 0.0729 0.0729 1.4 0.1376 0.9030 0.9030
−1.5 0.1204 0.0841 0.0841 1.5 0.1204 0.9159 0.9159
−1.4 0.1376 0.0970 0.0970 1.6 0.1049 0.9271 0.9271
−1.3 0.1565 0.1117 0.1117 1.7 0.0912 0.9369 0.9369
−1.2 0.1771 0.1284 0.1284 1.8 0.0790 0.9454 0.9454
−1.1 0.1993 0.1472 0.1472 1.9 0.0684 0.9528 0.9528
−1.0 0.2228 0.1683 0.1683 2.0 0.0590 0.9591 0.9591
−0.9 0.2471 0.1917 0.1917 2.1 0.0509 0.9646 0.9646
−0.8 0.2719 0.2177 0.2177 2.2 0.0439 0.9694 0.9694
−0.7 0.2965 0.2461 0.2461 2.3 0.0378 0.9734 0.9734
−0.6 0.3202 0.2770 0.2770 2.4 0.0326 0.9769 0.9769
−0.5 0.3422 0.3101 0.3101 2.5 0.0280 0.9800 0.9800
−0.4 0.3616 0.3453 0.3453 2.6 0.0242 0.9826 0.9826
−0.3 0.3777 0.3823 0.3823 2.7 0.0208 0.9848 0.9848
−0.2 0.3897 0.4207 0.4207 2.8 0.0180 0.9868 0.9868
−0.1 0.3972 0.4601 0.4601 2.9 0.0155 0.9884 0.9884


Note: QPVII(x)[1]QPVIIx1 : Gauss–Laguerre integration; QPVII(x)[2]QPVIIx2: Kotz and Nadarajah (2001).


Its bivariate copula density can be given as follows:


cPVIIuv=qPVIIQPVII−1uQPVII−1vqp7Q2−1uqp7Q2−1v=ΓN−1ΓNΓN−1221+x2mN−121+y2mN−121−ρ21+x2+y2−2ρxym1−ρ2N(7.74)

where x=Qp7−1u,y=Qp7−1v.



Marginal CDF of Symmetric Pearson Type II Distribution

Again, based on Fang et al. (2002), the probability density function of symmetric bivariate Pearson II distribution (for z = [x, y]z=xy) can be given as follows:


qPIIz=qPIIxy=m+1π1−ρ21−x2+y2−2ρxy1−ρ2m,xyR−1xyT≤1;m>−10,otherwise(7.75)

The marginal CDF of symmetric Pearson type II distribution can be expressed as follows:


Qp2x=Γm+2πΓm+32∫−1×1−y2m+12dy;x≤1(7.76)

Applying the Gauss–Legendre integration method, we can compute the marginal CDF of the symmetric Pearson type II distribution using the following:


∫abfxdx=b−a2∫−11fb−a2ξ+b+a2dξ≈b−a2∑k=1nwxkfb−a2xk+b+a2(7.77)

Table 7.4 lists the abscissa and the weight of the Gauss–Legendre integration.




Table 7.4. Abscissa and weight of the Gauss–Legendre integration.


































































































































No

k		 Abscissa

xkxk		 Weight

ω(xk)ωxk		 No

K		 Abscissa

xkxk		 Weight

ω(xk)ωxk
1 0.99726 0.007018 17 0.048308 0.09654
2 −0.98561 0.016277 18 0.144472 0.095638
3 −0.96476 0.025391 19 0.239287 0.093844
4 −0.93491 0.034275 20 0.331869 0.091174
5 −0.89632 0.042836 21 0.421351 0.087652
6 −0.84937 0.050998 22 0.5069 0.083312
7 −0.79448 0.058684 23 0.587716 0.078194
8 −0.73218 0.065822 24 0.663044 0.072346
9 −0.66304 0.072346 25 0.732182 0.065822
10 −0.58772 0.078194 26 0.794484 0.058684
11 −0.5069 0.083312 27 0.849368 0.050998
12 −0.42135 0.087652 28 0.896321 0.042836
13 −0.33187 0.091174 29 0.934906 0.034275
14 −0.23929 0.093844 30 0.964762 0.025391
15 −0.14447 0.095638 31 0.985612 0.016277
16 −0.04831 0.09654 32 0.997264 0.007018

Similar to the marginal CDF of the symmetric Pearson Type VII distribution, the marginal CDF of symmetric Pearson type II distribution may be solved using the incomplete beta function as follows:


Qp2x=12I1−x2m+3212,−1≤x≤01−12I1−x2m+3212,0<x≤1(7.78)

Comparing the equation of incomplete beta function given by Kotz and Nadarajah (2001), the marginal CDFs computed from the two methods with the given parameter m are very close, as shown in Table 7.5.




Table 7.5. Marginal CDF of symmetric Pearson type II distribution (m = 4.5).
















































































































xx qp2(x)qp2x Qp2(x)Qp2x[1] Qp2(x)Qp2x[2] xx qp2(x)qp2x Qp2(x)Qp2x[1] Qp2(x)Qp2x[2]
–1.0 0.0000 0.0000 0.0000 0.0 1.3535 0.5000 0.5000
–0.9 0.0003 0.0000 0.0000 0.1 1.2872 0.6331 0.6331
–0.8 0.0082 0.0003 0.0003 0.2 1.1036 0.7535 0.7535
–0.7 0.0467 0.0027 0.0027 0.3 0.8446 0.8513 0.8513
–0.6 0.1453 0.0117 0.0117 0.4 0.5661 0.9218 0.9218
–0.5 0.3212 0.0343 0.0343 0.5 0.3212 0.9657 0.9657
–0.4 0.5661 0.0782 0.0782 0.6 0.1453 0.9883 0.9883
–0.3 0.8446 0.1487 0.1487 0.7 0.0467 0.9973 0.9973
–0.2 1.1036 0.2465 0.2465 0.8 0.0082 0.9997 0.9997
–0.1 1.2872 0.3669 0.3669 0.9 0.0003 1.0000 1.0000


Note: Qp2(x)Qp2x[1]: Gauss–Legendre integration; Qp2(x)Qp2x[2]: Kotz and Nadarajah (2001).



7.3.2 Parameter Estimation


Generally speaking, the pseudo-maximum likelihood method may still be used to estimate parameters of meta-elliptical copulas (Nadarajah and Kotz, 2005). Here we will first introduce the pseudo-maximum likelihood function for Kotz and Pearson type meta-elliptical copulas. Then, we again focus on meta-Gaussian and meta-Student t copulas with examples.



Bivariate Symmetric Kotz Type Distribution

The joint probability density function of the bivariate symmetric Kotz type distribution can be given as follows:


fxy=srNsx2+y2−2ρxyN−1πΓNs1−ρ2N−12exp−rx2+y2−2ρxy1−ρ2s(7.79)

Then, the log-likelihood function can be given as follows:


logLNrsρ=lns+Nlogrs−lnπ−lnΓ(Ns)+12−Nln1−ρ2+N−1lnx2+y2−2ρxy−rx2+y2−2ρxy1−ρ2s(7.79a)

Taking the first-order derivative of Equation (7.79a) with respect to parameters N, r, s, ρN,r,s,ρ, we have the following:


∂logL∂N=logrs−1sΨNs−ln1−ρ2+lnx2+y2−2ρxy∂logL∂r=Nrs−x2+y2−2ρxy1−ρ2s∂logL∂s=1s−Nlnrs2+Ns2ΨNs−rx2+y2−2ρxy1−ρ2slnx2+y2−2ρxy1−ρ2∂logL∂ρ=2N−1ρ21−ρ2−2N−1xyx2+y2−2ρxy−2rsρx2+y2—1+ρ2xy1−ρ22×2+y2−2ρxy1−ρ2s−1(7.79b)


Bivariate Pearson Type VII Distribution

The log-likelihood function of the bivariate Pearson type VII distribution [Equation (7.71)] can be written as:


logLNmρ=lnN−1−lnπm−12ln1−ρ2−Nln1+x2+y2−2ρxym1−ρ2(7.80)

Taking the first-order derivative of Equation (7.80) with respect to parameters N, m, ρN,m,ρ, we have the following:


∂logL∂N=1N−1−ln1+x2+y2−2ρxym1−ρ2∂logL∂m=Nx2+y2−2ρxym21−ρ21+x2+y2−2ρxym1−ρ2−1−1m∂logL∂ρ=ρ1−ρ2−2Nρx2+y2−1+ρ2xym1−ρ21+x2+y2−2ρxym1−ρ2−1(7.80a)


Bivariate Pearson Type II Distribution

The log-likelihood function of the bivariate Pearson type II distribution (Equation (7.75)) can be written as follows:


logLmρ=lnm+1−lnπ−12ln1−ρ2+mln1−x2+y2−2ρxy1−ρ2(7.81)

Taking the first-order derivative of Equation (7.81) with respect to parameters m, ρm,ρ, we have the following:


∂logL∂m=1m+1+ln1−x2+y2−2ρxy1−ρ2∂logL∂ρ=ρ1−ρ2−2mρx2+y2−1+ρ2xy1−rho221−x2+y2−2ρxy1−ρ2−1(7.81a)

Setting Equations (7.79b), (7.80a), and (7.81a) to 0, we can estimate the parameters of the bivariate Kotz, Pearson VII, and Pearson II distributions by solving these equations simultaneously.




Example 7.12 Estimation of parameters of meta-Gaussian copula with the data given in Table 7.6.




Table 7.6. Three-dimensional data sample.























































































































































































































































No. u1u1 u2u2 u3u3 No. u1u1 u2u2 u3u3
1 0.8085 0.4026 0.7069 26 0.8044 0.3380 0.9206
2 0.8845 0.9449 0.9775 27 0.8441 0.3217 0.7441
3 0.0483 0.0201 0.0259 28 0.3713 0.5469 0.3967
4 0.5818 0.4478 0.7189 29 0.8165 0.5460 0.6650
5 0.7066 0.6085 0.6556 30 0.0444 0.2351 0.2073
6 0.0543 0.5992 0.0555 31 0.6413 0.8358 0.7090
7 0.4799 0.3308 0.4113 32 0.0675 0.2407 0.1012
8 0.7468 0.6777 0.6236 33 0.0142 0.1737 0.0638
9 0.9989 0.9913 0.9984 34 0.3875 0.7339 0.1912
10 0.9353 0.9649 0.9661 35 0.0237 0.0136 0.0091
11 0.0002 0.0012 0.0033 36 0.7743 0.8217 0.8119
12 0.9388 0.9533 0.9835 37 0.2967 0.8092 0.6397
13 0.8777 0.7798 0.7347 38 0.5267 0.2084 0.1927
14 0.2764 0.7564 0.4758 39 0.8736 0.8376 0.8816
15 0.8212 0.8777 0.7088 40 0.0968 0.0587 0.0861
16 0.4701 0.4711 0.4284 41 0.4120 0.0877 0.4317
17 0.7744 0.1112 0.4433 42 0.3236 0.4496 0.3733
18 0.4937 0.6475 0.8518 43 0.2043 0.7927 0.6416
19 0.7424 0.5635 0.8267 44 0.5628 0.9067 0.5870
20 0.7120 0.9838 0.9100 45 0.1844 0.2117 0.2585
21 0.9757 0.5134 0.7641 46 0.2724 0.5463 0.4876
22 0.3326 0.3769 0.1272 47 0.0737 0.3664 0.3733
23 0.8493 0.6129 0.7113 48 0.5192 0.2766 0.6553
24 0.7328 0.9191 0.9038 49 0.3644 0.6738 0.8504
25 0.5228 0.4322 0.6576 50 0.9005 0.3035 0.8588

Solution: Let {x1i, x2i, …, xdi}x1ix2i…xdi be a d-dimensional sample where i = 1, …, n, u1i = F1(x1i), …, udi = Fd(xdi)i=1,…,n,u1i=F1x1i,…,udi=Fdxdi. The parameter space is denoted as θ = {Σ : Σ ∈ Ω}θ=Σ:Σ∈Ω, where ΣΣ is symmetric and a positive definite matrix. Applying Equation (7.39a), the log-likelihood function of the d-dimensional meta-Gaussian copula can be written as follows:


logLθ=logLΣ=−n2lnΣ−12∑i=1nξiTΣ−1−Iξi

=−N2lnΣ−12trΣ−1∑i=1nξiTξi(7.82)

where ξi = [x1i, …, xdi]T = [Φ−1(u1i), …., Φ−1(udi)]Tξi=x1i…xdiT=Φ−1u1i….Φ−1udiT; tr(⋅)tr⋅ trace of the matrix.


Assuming Equation (7.82) is differentiable in θθ, parameters of the meta-Gaussian copula can be solved for by ∂logL∂θ=0 as follows:


∂logL∂Σ=−12tr−(Σ−12∑i=1nξiTξi)=−12trΣ−1nId−Σ−1∑i=1nξiTξi=0(7.82a)

From Equation (7.82a), we have the following:


nId−Σ−1∑i=1nξiTξi=0⇒Σ̂=1n∑i=1nξiTξi(7.82b)

To estimate the parameters (i.e., covariance matrix) of the meta-Gaussian copula, we first need to compute ξi = [Φ−1(u1i), Φ−1(u2i), Φ−1(u3i)]; Φ(⋅) : inverse of N(0, 1),ξi=Φ−1u1iΦ−1u2iΦ−1u3i;Φ⋅:inverse ofN01, as shown in Table 7.7.




Table 7.7. Inverse normal distribution: N(0,1).




































































































































































































































































































































































































































No. u1u1 u2u2 u3u3 Φ−1(u1)Φ−1u1 Φ−1(u2)Φ−1u2 Φ−1(u3)Φ−1u3
1 0.8085 0.4026 0.7069 0.8723 –0.2466 0.5444
2 0.8845 0.9449 0.9775 1.1977 1.5976 2.0040
3 0.0483 0.0201 0.0259 –1.6612 –2.0523 –1.9443
4 0.5818 0.4478 0.7189 0.2064 –0.1311 0.5797
5 0.7066 0.6085 0.6556 0.5435 0.2754 0.4004
6 0.0543 0.5992 0.0555 –1.6046 0.2512 –1.5941
7 0.4799 0.3308 0.4113 –0.0504 –0.4378 –0.2242
8 0.7468 0.6777 0.6236 0.6643 0.4613 0.3149
9 0.9989 0.9913 0.9984 3.0622 2.3775 2.9566
10 0.9353 0.9649 0.9661 1.5163 1.8109 1.8264
11 0.0002 0.0012 0.0033 –3.4847 –3.0458 –2.7162
12 0.9388 0.9533 0.9835 1.5446 1.6775 2.1332
13 0.8777 0.7798 0.7347 1.1638 0.7714 0.6271
14 0.2764 0.7564 0.4758 –0.5935 0.6947 –0.0606
15 0.8212 0.8777 0.7088 0.9201 1.1633 0.5499
16 0.4701 0.4711 0.4284 –0.0750 –0.0724 –0.1805
17 0.7744 0.1112 0.4433 0.7533 –1.2202 –0.1427
18 0.4937 0.6475 0.8518 -0.0158 0.3787 1.0442
19 0.7424 0.5635 0.8267 0.6508 0.1598 0.9411
20 0.7120 0.9838 0.9100 0.5593 2.1394 1.3409
21 0.9757 0.5134 0.7641 1.9729 0.0335 0.7197
22 0.3326 0.3769 0.1272 –0.4328 –0.3137 –1.1396
23 0.8493 0.6129 0.7113 1.0332 0.2869 0.5573
24 0.7328 0.9191 0.9038 0.6213 1.3990 1.3035
25 0.5228 0.4322 0.6576 0.0573 –0.1707 0.4060
26 0.8044 0.3380 0.9206 0.8574 –0.4179 1.4092
27 0.8441 0.3217 0.7441 1.0114 –0.4630 0.6561
28 0.3713 0.5469 0.3967 –0.3284 0.1179 –0.2619
29 0.8165 0.5460 0.6650 0.9022 0.1157 0.4262
30 0.0444 0.2351 0.2073 –1.7019 –0.7221 –0.8160
31 0.6413 0.8358 0.7090 0.3620 0.9773 0.5504
32 0.0675 0.2407 0.1012 –1.4947 –0.7041 –1.2746
33 0.0142 0.1737 0.0638 –2.1916 –0.9396 –1.5240
34 0.3875 0.7339 0.1912 –0.2859 0.6248 –0.8735
35 0.0237 0.0136 0.0091 –1.9832 –2.2086 –2.3609
36 0.7743 0.8217 0.8119 0.7532 0.9220 0.8849
37 0.2967 0.8092 0.6397 –0.5340 0.8749 0.3576
38 0.5267 0.2084 0.1927 0.0670 –0.8120 –0.8681
39 0.8736 0.8376 0.8816 1.1436 0.9848 1.1832
40 0.0968 0.0587 0.0861 –1.2999 –1.5657 –1.3654
41 0.4120 0.0877 0.4317 –0.2224 –1.3549 –0.1719
42 0.3236 0.4496 0.3733 –0.4576 –0.1266 –0.3232
43 0.2043 0.7927 0.6416 –0.8265 0.8157 0.3627
44 0.5628 0.9067 0.5870 0.1580 1.3205 0.2199
45 0.1844 0.2117 0.2585 –0.8989 –0.8005 –0.6479
46 0.2724 0.5463 0.4876 –0.6056 0.1163 –0.0310
47 0.0737 0.3664 0.3733 –1.4490 –0.3413 –0.3231
48 0.5192 0.2766 0.6553 0.0483 –0.5928 0.3996
49 0.3644 0.6738 0.8504 –0.3468 0.4504 1.0383
50 0.9005 0.3035 0.8588 1.2842 –0.5143 1.0749

Applying Equation (7.82b), we have Σ=10.67000.87580.670010.79450.87580.79451.




Example 7.13 Show how to estimate parameters of the meta-Student t copula.


Let {x1i, x2i, …, xdi}x1ix2i…xdi be a d-dimensional sample where i = 1, …, n, u1i = F1(x1i), …, udi = Fd(xdi)i=1,…,n,u1i=F1x1i,…,udi=Fdxdi. In the case of meta-Student t copula, its parameter space is θ = {(ν, Σ) : ν ∈ (1, ∞), Σ ∈ Ω }θ=νΣ:ν∈1∞Σ∈Ω. In the same way as in the meta-Gaussian copula, ΣΣ is symmetric and positive definite. Applying the meta-Student t copula density function (i.e., Equation (7.57)), the log-likelihood function can be given as follows:


logLνΣ=nlnΓν+d2Γν+12+nd−1lnΓν2Γν+12−n2lnΣ−ν+d2∑i=1nln1+ξiTΣ−1ξiν+ν+12∑i=1n∑j=1dln1+ξji2ν(7.83)

where ξi=Tν−1u1i…Tν−1udiT, and νν is the degree of freedom.


To estimate the fitted parameters θ̂=ν̂Σ̂, we may apply the following two approaches:




  1. 1. Optimizing the log-likelihood function (Equation (7.83)) numerically with the constraint of ΣΣ being symmetric and with ones on the main diagonal. With this constraint, the MLE estimate of Σ̂ may not be positive and semidefinite.



  2. 2. Estimate Σ̂ and νν separately.




    • Σ̂ may be estimated from the sample Kendall tau using the following:


      τ̂UiUj=2πarcsinρ̂ij⇒ρ̂ij=sinπ2τ̂ij(7.84)


      where τ̂ij=τ̂UiUj is the sample Kendall tau between random variable UiUi and UjUj; and ρ̂ij is the off-diagonal element of correlation matrix ΣΣ. In the same way as in approach 1, the estimated correlation matrix may not be positive definite.



    • Estimate the single parameter νν using MLE (Equation (7.83)) by fixing Σ̂.



For the estimated Σ̂ not being positive and semidefinite, we can apply the procedure discussed by McNeil et al. (2005) to convert it into positive definite matrix with the procedure as follows:




  1. i. Compute the eigenvalue decomposition Σ = EDETΣ=EDET, where E is an orthogonal matrix that contains eigenvectors, and D is the diagonal matrix that contains all the eigenvalues.



  2. ii. Construct a diagonal matrix D˜ by replacing all negative eigenvalues in D by a small value δ>0δ>0.



  3. iii. Compute Σ˜=ED˜ET, Σ˜ is positive definite but not necessarily a correlation matrix.



  4. iv. Apply the normalizing operator P to obtain the desired correlation matrix.


Specifically, for the bivariate case, the parameters that need to be estimated are θ = (ρ, ν)θ=ρν. Thus, the log-likelihood function (i.e., Equation (7.83)) can be rewritten as follows:


logL(ρ,ν)=nlnΓ(ν+22)+nlnΓ(ν2)−2nlnΓ(ν+12)−n2ln(1−ρ2)−ν+22∑i=1nIn(1+(ξ1i2−2ρξ1iξ2i+ξ2i2)(1−ρ2)ν  )+ν+12∑i=1n∑j=12ln(1+ξji2ν)(7.85)

Taking the first-order derivative with respect to ρ, ν,ρ,ν, we have the following:


∂logLρν∂ρ=nρ1−ρ2+ν+21−ρ2∑i=1nξ1i−ρξ2iξ2i−ρξ1iν1−ρ2+ξ1i2−2ρξ1iξ2i+ξ2i2∂logLρν∂ρ=−n2Ψν+22+n2Ψν2−nΨν+12−12∑i=1nln1+ξ1i2−2ρξ1iξ2i+ξ2i2ν1−ρ2(7.85a)

+ν+22ν2∑i=1nνξ1i2−2ρξ1iξ2i+ξ2i2ν1−ρ2+ξ1i2−2ρξ1iξ2i+ξ2i2+12∑i=1n∑j=12ln1+ξji2ν−ν+12ν2∑i=1n∑j=12ξji21+ξji2ν−ν+2∑i=1nξ1idξ1idν+ξ2idξ2idν−ρξ1idξ2idν+ξ2idξ1idνν1−ρ2+ξ1i2−2ρξ1iξ2i+ξ2i2+ν+1∑i=1n∑j=12ξjidξjidνν+ξji2(7.85b)



Example 7.14 Using the data given in Table 7.7, estimate the parameters for the bivariate (using u1, u2)u1,u2) and the trivariate meta-Student t copula.


Solution:


Bivariate meta-Student t copula (using u1, u2)u1,u2)




  • Approach 1


    For the bivariate case, we will apply Equation (7.85), i.e., maximizing the bivariate meta-Student t log-likelihood function.


    The initial correlation coefficient is set as the sample correlation coefficient computed from the sample Kendall tau (τ̂0=0.3812) using Equation (7.84) as follows:


    ρ̂0=sinπ2τ̂0=sin0.3812π2=0.5637.


    The initial degree of freedom (d.f.) is set as the lower limit (i.e., ν̂0=10).


    Then, the final parameter set θ̂=ρ̂ν̂:ρ∈−11ν>1 may be estimated using the optimization toolbox (e.g., the fmincon function) by minimizing the negative log-likelihood function (the objective function), which is the dual problem of the MLE estimation. We have the following:


    θ̂=ρ̂ν̂=0.55916.2531. With the estimated correlation coefficient, the correlation matrix is given as follows: Σ=10.55910.55911. The eigenvalue of the correlation matrix is λ=0.44091.5591, i.e., the correlation matrix is positive definite.


    Furthermore, one can use the MATLAB function copulafit to estimate the parameters of the meta-Student t copula using the MLE method. The function is given as follows:


    MLE: Σ̂ν̂=copulafit’t’data


    Using MLE from MATLAB, we have the following:


    Σ̂=10.55910.55911,ν̂=6.2542.



  • Approach 2


    Fixing ρ̂=0.5637, we haveν̂=6.4110.


Trivariate meta-Student t copula




  • Approach 1


    It is shown for the bivariate case that the parameters estimated using the embedded MATLAB function and those estimated using the fmincon by writing our own objective function are almost the same. So for the trivariate example, we will only show the results obtained from the embedded MATLAB function.


    Applying approach 1 and maximizing the log-likelihood function of the trivariate meta-Student t copula, using the embedded MATLAB function mentioned previously, we have the following ML method:


    Σ̂=10.58310.81710.583110.75180.81710.75181,ν̂=12.8139



  • Approach 2


    To apply approach 2, we first need to compute the sample correlation matrix from the sample Kendall tau using Equation (7.84) as follows:


    τ=10.38120.65880.381210.51020.65880.51021,Σ=10.56370.85980.563710.71830.85980.71831.


    The eigenvalue vector of ΣΣ is computed as λ = [0.1116, 0.4537, 2.4347]Tλ=0.11160.45372.4347T. Thus, we reach the conclusion that the correlation matrix is positive definite.


    Fixing the correlation matrix ΣΣ, we only have one parameter, i.e., νν, that needs to be estimated. Optimizing the log-likelihood equation (i.e., Equation (7.83)), we can estimate νν with an initial estimate of ν̂0=2. Using the fmincon function, we have ν̂=20.4038.


    It should be noted here for the meta-Student t copula that one can also use the following embedded function: Σ̂ν̂=copulafit’t’data,’Method’,’ApproximateML’. This estimation method is considered as a good estimation only if the sample size is large enough.



7.4 Summary


In this chapter, we have summarized and discussed the properties of meta-elliptical copulas. We have explained the procedures on how to construct and apply the meta-elliptical copulas, especially for the meta-Gaussian and meta-Student t copulas. Comparing meta-Gaussian and meta-Student t copulas, both copulas may be applied to model the dependence of entire range. The Student t copula possesses the symmetric upper (lower) tail dependence, while the meta-Gaussian copula does not possess the tail dependence. The meta-elliptical copula may be applied for the multivariate frequency analysis.




References


Fang, H. B., Fang K. T., and Kotz, S. (2002). The meta-elliptical distributions with given marginals. Journal of Multivariate Analysis, 82, 1–16.

Genest, C., Favre, A. C., Be´liveau, J., and Jacques, C. (2007). Meta-elliptical copulas and their use in frequency analysis of multivariate hydrological data. Water Resources Research, 43, W09401, doi:10.1029/2006WR005275.

Joe, H. (1997). Multivariate Models and Dependence Concept. Chapman & Hall, New York.

Kotz, S. and Nadarajah, S. (2001). Some extreme type elliptical distributions. Statistics & Probability Letters, 54, 171–182.

McNeil, A., Frey, R., and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton: Princeton University Press.

Nadarajah, S. ( 2006). Fisher information for the elliptically symmetric Pearson distributions. Applied Mathematics and Computation, 178, 195–206.

Nadarajah, S. (2007). A bivariate gamma model for drought. Water Resources Research, 43, W08501, doi:10.1029/2006WR005641.

Nadarajah, S. and Kotz, S. (2005). Information matrices for some elliptically symmetric distribution. SORT, 29(1), 43–56.

Zhang, G. (2000). Multiple complex Gauss–Legendre integral formulae and application. Journal of Lanzhou University (Natural Sciences), 36(5), 30–34.

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