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The goal of this paper is to introduce new families of multivariate copulas, extending the chi-square copulas, the Fisher copula, and squared copulas. The new families are constructed from existing copulas by first transforming their margins to standard Gaussian distributions, then transforming these variables into non-central chi-square variables with one degree of freedom, and finally by considering the copula associated with these new variables. It is shown that by varying the non-centrality parameters, one can model non-monotonic dependence, and when one or many non-centrality parameters are outside a given hyper-rectangle, then the copula is almost the same as the one when these parameters are infinite. For these new families, the tail behavior, the monotonicity of dependence measures such as Kendall's tau and Spearman's rho are investigated, and estimation is discussed. The R package NCSCopula [7] can be used to estimate the parameters for several copula families.
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On non-central squared copulas
Bouchra R. Nasri1,
Abstract
The goal of this paper is to introduce new families of multivariate copulas, extending the chi-square copulas, the Fisher
copula, and squared copulas. The new families are constructed from existing copulas by first transforming their
margins to standard Gaussian distributions, then transforming these variables into non-central chi-square variables
with one degree of freedom, and finally by considering the copula associated with these new variables. It is shown
that by varying the non-centrality parameters, one can model non-monotonic dependence, and when one or many
non-centrality parameters are outside a given hyper-rectangle, then the copula is almost the same as the one when
these parameters are infinite. For these new families, the tail behavior, the monotonicity of dependence measures such
as Kendall’s tau and Spearman’s rho are investigated, and estimation is discussed. The R package NCSCopula [7] can
be used to estimate the parameters for several copula families.
Keywords: Chi-square copulas, Fisher copula, squared copulas, non-centrality parameters, tail behavior, estimation,
non-monotonic dependence.
1. Introduction
Over the last decades, Archimedean copulas and Elliptical copulas have received a lot of attention. In fact, in most
articles, when comparisons or simulations are made, the Gaussian, Student, Clayton, Frank, and Gumbel copulas are
often the only one used. Even if these families and their rotations have interesting properties, they have been criticized
by Dette et al. [4] since they all have a monotonic dependence structure. More precisely, as defined in Nelsen [9,
p.196], these copulas are all stochastically increasing or stochastically decreasing, meaning that if (U,V)C, then
for all v[0,1], P(Vv|U=u) is non-decreasing (resp. non-increasing ) in u(0,1). In particular, all the
copula families considered by Dette et al. [4], namely Elliptical and Archimedean copula and their rotations, cannot
reproduce the non-monotonic dependence structure between Xand Y, where Y=(X0.5)2+, with XU(0,1)
is independent of N(0, σ2), and σ=0.1. However, Nasri et al. [8] showed that the chi-square copula family
introduced by B´
ardossy [1] and explored more in depth in Quessy et al. [12], was able to capture the non-monotonic
dependence between Xand Y.
In the bivariate case, the chi-square copula is the unique copula associated with the random variables (Z1+a1)2
and (Z2+a2)2, where Z1and Z2are joint standard Gaussian variables with correlation ρ. In fact, assuming a1,a2>0,
the associated copula is the same as the copula of the random variables Z1+Z2
1
2a1and Z2+Z2
2
2a2. One obtains the Gaussian
copula as a limiting case by letting a1→ ∞ and a2→ ∞. The chi-square copula can indeed reproduce non-monotonic
Preprint submitted to Statistics and Probability Letters January 13, 2020
dependence as can be seen by letting only a2→ ∞, since the limiting copula is then the copula associated with
(Z1+a1)2and Z2. Quessy et al. [12] examined some properties of the chi-square copula for general non-centrality
parameters. In particular they give an explicit expression for Kendall’s tau. They also studied the tail behavior when
a1=a2=0. As for estimation and modeling, they took a1=a2. Estimation issues were not discussed. However,
as illustrated in column (c) of Figure 1, one can see that the pseudo log-likelihood is flat on the interval a[3,6],
while the real value of the parameters is a1=a2=4. This shows that the chi-square copula converges quite fast
to the Gaussian copula when a=a1=a2is large, so the range of parameters must be bounded. Since the work of
Quessy et al. [12], other copula families based on chi-square variables were defined. The Fisher copula introduced
in Favre et al. [5] can be defined as the copula between Z2
1and Z2
2, where Z1=Φ1(U1) and Z2=Φ1(U2), where
(U1,U2) follows the Student copula and Φis the cumulative distribution function (cdf) of the standard Gaussian. This
method of construction was extended in Quessy and Durocher [11] to any other copula, defining was might be called
squared copulas. In these cases, if (U1,U2)C, the new copula is the joint distribution of (|2U11|,|2U21|).
Tail indexes were computed and estimation of the parameters of the copula Cwere studied. Since the new copulas in
Favre et al. [5], Quessy and Durocher [11] are based on central chi-square variable, they lack interesting properties
such as non-monotonic dependence structure.
In this paper, ones studies some properties of non-central squared copulas, which are constructed from non-
central chi-square variables. As particular cases, one obtains all the copulas in Favre et al. [5], Quessy and Durocher
[11], Quessy et al. [12]. More precisely, the new copula families are defined in Section 2, together with expressions for
the copula and its density. Next, the limiting behavior and the estimation are discussed in Section 3. The monotonicity
of dependence measures such as Kendall’s tau and Spearman’s rho is investigated in Section 4. Finally, tail dependence
is studied in Section 5, solving the conjecture of Quessy et al. [12] for the chi-square copula. Auxiliary results and the
proofs of the main results are given in the Appendix.
2. Non-central squared copulas
In this section, ones defines the new families of copulas and one studies their asymptotic behavior. To this end, for
a given copula C, let (U1,...,Ud)C, and set Z1=Φ1(U1),...,Zd=Φ1(Ud) be the normal transforms. The non-
central squared copula ˜
Ca,a=(a1,...,ad)[0,)d, is the copula associated with the random vector of non-central
chi-square variables (Z1+a1)2,...,(Zd+ad)2. It is the also the same one as the copula of (|Z1+a1|,...,|Zd+ad|).
Here, without loss of generality, the non-centrality parameters are restricted to be non-negative. In fact, if aj>0,
then (Zjaj)2=(Zj+aj)2, so Zj=Φ(1 Uj), leading to multivariate rotations of the original copula C. In
fact, in the bivariate case, ˜
Ca1,a2=˜
C90,a1,a2,˜
Ca1,a2=˜
C180,a1,a2, and ˜
Ca1,a2=˜
C270,a1,a2, where the rotations of Care
C90(u1,u2)=u2C(1 u1,u2), which is the copula of (1 U1,U2), C180(u1,u2)=u1+u21+C(1 u1,1u2),
which is the copula of (1 U1,1U2), and C270(u1,u2)=u1C(u1,1u2), which is the copula of (U1,1U2). See,
e.g., Brechmann and Schepsmeier [2]. Further note that ˜
C0corresponds to the copula defined in Quessy and Durocher
2
[11].
Here, one uses almost the same notations as in Quessy et al. [12]. For any u(1,1), define ˜
ha(u)=Φ{ha(u)},
where ha(u)=sign(u)G1
a(|u|)a, and Ga(x)=P{|Z+a| ≤ x}=Φ(xa)Φ(xa),x0. It follows that the
non-central squared copula ˜
Cais given by
˜
Ca(u1,...,ud)=Phd
j=1{˜
haj(uj)<Uj˜
haj(uj)}i=X
(1,...,d)∈{−1,1}d
d
Y
j=1
j
Cn˜
ha1(1u1),...,˜
had(dud)o.(1)
Next, to compute the density ˜caof ˜
Ca, note that for any ∈ {−1,1}, and u(0,1), d
du n˜
ha(u)o=φ{ha(u)}
φ{ha(u)}+φ{ha(u)},
where φ=Φ0is the density of the standard Gaussian distribution. As a result, if cis the density of the copula C, then
the density ˜caof ˜
Cais given by
˜ca(u1,...,ud)=X
(1,...,d)∈{−1,1}d
d
Y
j=1
φ{haj(juj)}
φ{haj(uj)}+φ{haj(uj)}
cn˜
ha1(1u1),...,˜
had(dud)o.(2)
Remark 1. Usually Archimedean copulas are not used for multivariate data since it imposes that all pairs have the
same distribution. This is not the case here for ˜
Caeven if Cis Archimedean, unless a1=··· =ad.
3. Limiting behavior and non-centrality parameters estimation
One now discusses the limiting behavior of ˜
Cawhen one or many parameters tends to infinity. To this end, suppose
that aj>0 for j∈ {1,...,k}. Then ˜
Cais the copula associated with the vector X=(X1,...,Xd), where Xj=Zj+Z2
j
2aj
for j∈ {1,...,k}, and Xj=(Zj+aj)2otherwise. As a result, if one lets a1, . . . ak→ ∞, then ˜
C,...,,ak+1,...,adis the
copula between Z1,...,Zkand (Zj+aj)2,j>k, showing that one can get quadratic dependence. In this case, using
(1), one gets that
˜
C,...,,ak+1,...,ad(u)=Pk
j=1{Ujuj} ∩d
j=k+1{˜
haj(uj)<Uj˜
haj(uj)}.
As a result, when a1, . . . ad→ ∞, then ˜
Ca˜
C=C. Columns (a) and (b) of Figure 1display a simulation of bivariate
non-central squared copulas when a1=a2=1.5 (Case 1) and when a1=2, a2=0.3 (Case 2) computed from the
Gaussian, Student, Clayton, Frank and Gumbel copulas with standard Gaussian margins and Kendall’s τ=0.75. One
can easily see that in Case 1, the dependence seems monotonic, while in Case 2 the non-monotonic dependence is
pretty obvious and seems converging to the copula between Z1and (Z2+a2)2. Next, one shows how fast ˜
Caconverges
when some non-centrality parameters go to infinity. For stating the new theorem, there is no loss of generality in
assuming that the first kdparameters are large, the general case being obtained as a permutation of the indices.
Theorem 1. Set a=(a1,...,ad)[0,)dand suppose that a jb for all j ∈ {1,...,k}, with 1kd. Then
sup
u[0,1]d˜
Ca(u)˜
C,...,,ak+1,...,ad(u)kΦ(b).(3)
3
-2 0 2 4
-2
0
2
4
Gaussian copula (a1=2, a2 = 0.3)
-2 0 2
-2
0
2
Gaussian copula (a1=a2=1.5)
0246
a
700
800
900
LL
Gaussian
-2 0 2 4
-5
0
5
Student copula (a1=2, a2 = 0.3)
-5 0 5
-5
0
5
Student copula (a1=a2=1.5)
0246
a
700
800
900
LL
Student
-5 0 5
-4
-2
0
2
Clayton copula (a1=2, a2 = 0.3)
-2 0 2
-2
0
2
4
Clayton copula (a1=a2=1.5)
0246
a
500
1000
LL
Clayton
-2 0 2
-4
-2
0
2
Frank copula (a1=2, a2 = 0.3)
-4 -2 0 2
-5
0
5Frank copula (a1=a2=1.5)
0246
a
0
100
200
LL
Frank
-2 0 2
-2
0
2
Gumbel copula (a1=2, a2 = 0.3)
-2 0 2
-2
0
2
Gumbel copula (a1=a2=1.5)
0246
a
800
900
1000
LL
Gumbel
(c)
(b)(a)
Figure 1: Scatterplots of random samples of size n=1000 from ˜
Ca1,a2with standard Gaussian margins when the copula Cis Gaussian, Student,
Clayton, Frank, and Gumbel in columns (a) and (b). In addition, the Student copula has ν=12 degrees of freedom, and all copulas have the same
Kendall’s tau (τ=0.75). The graph of the pseudo log-likelihood for random samples of size n=1000 from ˜
Ca,aas a function of a(0,6] are
displayed in column (c), where the true values of the non-centrality parameters are a1=a2=4.
In particular, if ajb, for all j ∈ {1,...,d}, then
sup
u[0,1]d˜
Ca1,...,ad(u)C(u)dΦ(b).(4)
For example, in the bivariate case, it follows from Theorem 1that when b=3, Φ(3) =0.0013, so
sup
u[0,1]d˜
Ca(u)˜
C,a2(u).0013, if a[3,)×[0,3], sup
u[0,1]d˜
Ca(u)˜
Ca1,(u).0013, if a[0,3] ×[3,), and
sup
u[0,1]d˜
Ca(u)C(u).0026,if a[3,)2. As a result of these three inequalities, parameters a=(a1,a2) should
be restricted to [0,3]2in order to insure that they are estimated accurately. The same interval should be considered in
the multivariate case. Furthermore, Theorem 1also explains the behavior of the pseudo log-likelihoods in column (c)
of Figure 1, where the latter become flat for the five copula families considered when a1=a2[3,6].
4
Remark 2. In the case of general distribution functions F1,...,Fd, with 0 <Fj(0) <1, the copula associated with
|F1
1(U1)+a1|, . . . |F1
1(U1)+a1|is the copula of V1,...,Vd, where Vj=|UjFj{−2ajF1
j(Uj)}|,j∈ {1,...,d}.
Therefore, even in the case of symmetric distributions, the copula depends on Fjwhenever aj,0. Note that in
Theorem 1, the righthand side of Equation (3) would be Pk
j=1Fj(bj), whenever ajbj.
To assess the performance of the pseudo log-likelihood estimators, one generated random bivariate samples of
size n∈ {250,500,1000}from ˜
Ca1,a2, where Cis a Gaussian, Student (ν=12), Clayton, Frank, and Gumbel with
Kendall’s tau τ∈ {0.25,0.5,0.75}, and (a1,a2)∈ {(0.5,1.5),(0.5,2.5),(1.0,2.0), (1.5,2.0),(1.5,2.5),(2.0,2.5)}. In
each of these 270 scenarios, N=1000 replications were performed. Parameters (a1,a2, τ) were estimated using
the pseudo maximum likelihood method proposed in Genest et al. [6] and Shih and Louis [13], and the relative bias
and root mean square errors in percentage were computed. Table 1displays the results for τ=0.5. The results for
τ∈ {0.25,0.75}are available in the supplementary material. One also considered the case a1=a2∈ {0.5,1.5,2.0,2.5}
and τ=0.5 for the same five families, partially studied in Quessy et al. [12] for the Gaussian copula. These results are
displayed in Table 2. From Table 1and the tables in the supplementary material, one can see that in general τis easy to
estimate even if when the sample size is small and the level of dependence is low. Also the precision of the estimation
of all parameters increases significantly as τincreases or as nincreases and do not seem to depend significantly on the
copula family, which is a good news. It is normal that for small values of τ, the estimation error and bias are larger
since for all families but the Student, τnear 0 means independence and then the copula does not depend on a1,a2.
For the Student copula which has an additional parameter, the non-centrality parameters do not aect the estimation
of ν. Note that based on the results of Section 4, there is no clear relationship between the non-centrality parameters
and τ. For τ=0.25, the error and bias are larger when the smallest non-centrality parameter is close to 0, i.e., when
a1=0.5. And the performance increases dramatically when both coecients are close to 3. This property is no longer
true when τis larger. Finally, based on Table 2, one can see that generally the precision increases when a=a1=a2
increases or as nincreases. Also, comparing Tables 1and 2, the bias and the precision generally seems slightly better
than when there are three parameters to estimate except for the case a=a1=a2=0.5.
Remark 3. When d>3, instead of maximizing the full pseudo-likelihood, it might be better to use the technique
proposed in Quessy and Durocher [11], namely to use a composite pairwise pseudo-likelihood. This consists in
maximizing the sum over all pairs of variables of all log-pseudo-likelihoods. See also Varin et al. [14] for a review on
composite likelihood and Oh and Patton [10] for another application to large dimensional copulas.
4. Dependence measures
In the bivariate case, we now investigate the behavior of two measures of dependence, namely Kendall’s tau and
Spearman’s rho in terms of the non-centrality parameters a1,a2. For a given bivariate copula D, recall that
τ(D)=1+4Z(0,1)2
D(u1,u2)dD(u1,u2)=1+4E{D(U1,U2)},(5)
5
Table 1: Relative RMSE and bias (in parenthesis) in percentage for the estimation of the parameters (a1,a2, τ) for ˜
Ca1,a2when the copula Cis
Gaussian, Clayton, Frank, and Gumbel family with τ=0.5.
(a1,a2)n=250 n=500 n=1000
a1a2τa1a2τa1a2τ
Gaussian
(0.5,1.5) 48.0(4.3) 35.5(0.1) 12.8(4.2) 23.6(4.9) 26.2(0.0) 8.8(2.3) 16.9(3.2) 17.1(1.1) 6.0(1.6)
(0.5,2.5) 35.7(2.1) 24.8(9.7) 9.7(3.1) 13.0(1.5) 17.9(2.3) 5.7(1.7) 9.1(0.7) 14.0(0.8) 3.7(0.7)
(1.0,2.0) 40.4(1.0) 35.2(21.9) 11.9(4.2) 24.8(0.3) 30.0(16.4) 7.5(2.1) 16.2(1.4) 25.7(12.5) 4.5(1.3)
(1.5,2.0) 45.2(21.4) 22.7(7.2) 7.2(1.7) 36.7(16.7) 19.0(6.3) 4.8(0.6) 27.2(10.4) 16.1(5.7) 3.3(0.6)
(1.5,2.5) 42.6(18.6) 31.8(26.8) 7.0(1.6) 37.2(16.7) 29.6(26.4) 5.0(0.7) 28.7(11.7) 28.4(25.7) 3.3(0.4)
(2.0,2.5) 37.4(23.0) 23.0(11.7) 6.8(2.0) 36.6(26.1) 19.8(9.4) 4.4(0.7) 36.6(27.4) 18.5(8.9) 3.1(0.3)
Clayton
(0.5,1.5) 19.5(0.4) 12.6(1.0) 8.9(0.5) 13.2(0.0) 7.2(0.2) 6.1(0.3) 9.2(0.4) 5.0(0.2) 4.3(0.3)
(0.5,2.5) 22.2(2.1) 13.9(1.2) 7.0(1.2) 10.7(1.1) 8.1(0.2) 4.8(0.4) 7.9(0.4) 4.6(0.2) 3.5(0.2)
(1.0,2.0) 12.8(2.7) 16.7(4.9) 7.8(2.4) 7.2(0.9) 9.6(0.9) 5.3(0.7) 5.3(0.0) 5.4(0.0) 3.7(0.2)
(1.5,2.0) 18.7(1.6) 21.8(17.0) 8.2(1.8) 13.6(1.0) 20.5(15.9) 5.7(0.9) 10.4(0.6) 20.1(15.4) 4.1(0.6)
(1.5,2.5) 17.0(1.8) 37.2(32.6) 8.1(2.4) 12.5(1.4) 38.4(34.3) 5.5(1.5) 10.7(1.5) 39.4(35.6) 4.0(1.3)
(2.0,2.5) 18.7(6.5) 22.8(19.2) 6.8(0.9) 14.8(5.0) 21.6(18.8) 4.8(0.5) 11.3(3.7) 21.3(19.2) 3.5(0.6)
Frank
(0.5,1.5) 47.4(2.2) 47.0(9.4) 12.9(3.8) 19.9(5.0) 45.2(10.7) 9.2(2.9) 13.4(3.1) 39.5(10.0) 6.3(1.6)
(0.5,2.5) 39.8(2.5) 38.2(24.4) 11.2(3.3) 15.8(3.2) 34.0(20.2) 7.6(2.4) 11.0(2.3) 30.7(17.0) 5.2(1.3)
(1.0,2.0) 69.3(23.3) 41.3(13.7) 13.0(4.5) 65.1(24.0) 38.7(13.6) 8.6(2.0) 53.5(17.1) 37.5(13.3) 6.6(1.6)
(1.5,2.0) 61.9(32.8) 35.7(2.2) 9.2(2.9) 58.8(32.8) 32.6(2.4) 6.1(1.5) 57.5(33.7) 30.2(2.3) 3.6(0.5)
(1.5,2.5) 61.4(32.6) 33.7(19.5) 8.8(2.6) 61.6(37.2) 30.8(16.4) 5.6(1.1) 60.1(38.5) 29.7(16.1) 3.6(0.2)
(2.0,2.5) 41.0(15.0) 29.8(14.0) 8.2(2.6) 40.8(21.6) 26.7(10.3) 5.1(1.3) 41.3(26.0) 23.9(8.0) 3.2(0.5)
Gumbel
(0.5,1.5) 35.0(4.8) 38.1(1.6) 10.9(3.7) 20.2(2.9) 34.6(4.0) 7.8(1.7) 12.6(1.3) 25.3(3.5) 5.3(0.9)
(0.5,2.5) 20.9(3.2) 30.4(15.9) 9.5(3.2) 14.9(1.0) 24.1(9.6) 6.3(2.2) 9.5(0.2) 19.9(5.0) 4.2(0.8)
(1.0,2.0) 65.6(17.1) 40.5(21.7) 12.6(4.3) 53.1(14.0) 36.8(19.9) 8.8(2.3) 36.9(8.5) 34.1(18.1) 5.9(1.5)
(1.5,2.0) 65.9(38.6) 30.2(4.7) 8.5(2.0) 57.9(33.3) 28.0(2.7) 5.5(0.8) 52.0(28.7) 25.3(2.2) 3.9(0.5)
(1.5,2.5) 65.4(37.3) 33.5(22.9) 8.4(2.4) 57.9(33.6) 30.9(21.8) 5.4(0.6) 51.2(28.3) 29.1(20.2) 3.6(0.4)
(2.0,2.5) 43.0(27.2) 27.3(14.1) 7.7(2.1) 43.0(32.1) 23.7(11.3) 4.8(0.9) 42.8(34.1) 22.2(10.7) 3.3(0.4)
Student ν=12
(0.5,1.5) 33.7(2.8) 31.3(0.8) 16.8(1.0) 23.3(3.3) 23.4(0.8) 10.3(0.3) 16.4(2.3) 15.9(1.3) 7.0(0.8)
(0.5,2.5) 23.3(0.8) 23.9(8.8) 13.5(0.7) 14.0(0.2) 16.8(2.6) 7.4(0.8) 9.6(1.2) 12.6(0.7) 4.7(1.1)
(1.0,2.0) 42.4(0.1) 34.4(17.9) 11.3(3.7) 24.8(1.3) 28.1(13.0) 7.1(2.1) 16.9(0.8) 23.3(7.7) 4.9(1.2)
(1.5,2.0) 48.7(22.6) 24.8(10.1) 7.9(1.7) 44.4(22.1) 21.2(11.2) 5.2(1.1) 41.5(21.5) 19.5(10.0) 3.7(0.5)
(1.5,2.5) 47.7(22.5) 34.0(29.2) 7.6(1.3) 44.8(23.3) 33.4(30.0) 5.2(0.8) 39.7(19.8) 32.1(28.8) 3.6(0.5)
(2.0,2.5) 35.8(22.8) 24.4(14.5) 6.6(1.4) 35.4(25.7) 22.1(13.8) 4.6(1.0) 33.9(25.0) 19.3(11.8) 3.4(0.5)
Estimation of ν
(0.5,1.5) 86.1(20.5) 77.5(20.2) 70.6(20.4)
(0.5,2.5) 92.3(37.3) 96.1(63.3) 98.9(77.6)
(1.0,2.0) 80.4(23.1) 69.4(15.5) 60.9(15.1)
(1.5,2.0) 75.7(23.3) 66.3(22.2) 54.3(14.0)
(1.5,2.5) 73.7(22.1) 65.0(17.3) 55.3(15.1)
(2.0,2.5) 75.3(20.1) 69.3(23.0) 57.9(16.6)
where (U1,U2)D. In addition, if (V1,V2) is an independent copy of (U1,U2), then
E{D(U1,U2)}=P{U1V1,U2V2}.(6)
Note also that Spearman’s rho is defined by
ρS(D)=12 Z(0,1)2{D(u1,u2)u1u2}du1du2=3+12E(U1U2).(7)
6
Table 2: Relative RMSE and bias (in parenthesis) in percentage for the estimation of the parameters a=a1=a2and τfor ˜
Ca,awhen the copula C
is Gaussian, Clayton, Frank, and Gumbel family with τ=0.5.
a n =250 n=500 n=1000
aτaτaτ
Gaussian
0.5 94.6(20.5) 14.6(1.3) 79.0(26.6) 11.7(1.6) 68.9(29.0) 9.1(2.5)
1.0 41.4(10.2) 12.8(4.2) 28.1(5.6) 9.0(2.4) 14.2(1.2) 5.3(0.9)
1.5 22.2(0.2) 7.9(2.0) 13.6(0.2) 5.3(0.9) 8.6(0.3) 3.6(0.6)
2.0 21.8(1.5) 6.7(1.9) 16.3(1.2) 4.5(0.8) 10.9(0.9) 3.0(0.4)
2.5 18.9(5.7) 6.5(1.9) 14.2(0.3) 4.4(0.8) 11.9(0.5) 3.0(0.6)
Clayton
0.5 50.1(11.6) 12.2(3.3) 30.5(4.3) 7.9(1.3) 19.7(3.4) 5.2(0.9)
1.0 21.8(1.1) 10.7(0.9) 15.3(0.3) 7.7(0.2) 11.0(0.1) 5.4(0.2)
1.5 15.1(0.1) 7.9(0.8) 9.3(0.4) 5.5(0.5) 6.5(0.1) 4.0(0.1)
2.0 16.9(2.7) 6.7(0.7) 10.1(0.6) 4.8(0.3) 6.1(0.1) 3.4(0.1)
2.5 15.4(1.8) 6.3(0.7) 11.4(1.1) 4.6(0.3) 7.9(1.1) 3.2(0.1)
Frank
0.5 130.7(82.7) 23.2(16.3) 92.1(60.5) 20.8(14.5) 68.9(43.2) 17.4(11.5)
1.0 41.9(4.3) 12.7(2.4) 28.8(0.6) 10.7(2.1) 19.0(1.9) 8.7(2.0)
1.5 39.7(8.4) 9.5(2.6) 32.6(7.7) 6.7(1.4) 23.5(4.5) 4.1(0.6)
2.0 32.3(0.4) 8.5(2.6) 29.0(4.4) 5.1(1.1) 27.4(7.0) 3.3(0.7)
2.5 30.8(16.0) 8.4(2.5) 27.1(12.7) 5.0(1.3) 23.6(9.2) 3.2(0.5)
Gumbel
0.5 98.2(43.5) 14.6(5.0) 59.4(23.6) 11.4(3.2) 42.8(15.3) 9.1(2.5)
1.0 44.4(0.4) 12.8(3.9) 29.2(3.3) 10.0(2.6) 20.2(3.1) 7.4(1.8)
1.5 30.4(1.8) 9.1(2.6) 21.6(1.2) 6.0(1.0) 12.7(0.2) 3.9(0.5)
2.0 25.1(1.8) 7.6(2.0) 22.1(3.8) 4.8(0.5) 18.5(3.1) 3.4(0.3)
2.5 25.6(12.4) 8.0(2.1) 19.4(5.0) 4.8(0.9) 15.6(1.0) 3.4(0.4)
Student ν=12
0.5 90.7(13.5) 27.8(7.6) 76.5(24.4) 18.3(2.0) 67.4(25.7) 10.2(1.1)
1.0 42.1(8.8) 13.1(2.2) 26.9(5.3) 8.9(2.1) 14.9(1.9) 5.4(1.1)
1.5 22.9(0.5) 7.8(2.0) 13.4(0.6) 5.2(0.9) 9.0(0.2) 3.8(0.4)
2.0 22.1(1.3) 6.9(1.4) 16.3(2.1) 4.6(0.6) 9.8(0.3) 3.3(0.3)
2.5 19.6(3.4) 6.7(1.1) 14.9(1.6) 4.5(0.8) 11.5(0.1) 3.1(0.4)
Estimation of ν
0.5 80.0(22.4) 73.4(22.7) 66.7(23.2)
1.0 70.4(21.5) 63.1(23.2) 51.1(17.0)
1.5 69.0(21.1) 62.1(19.1) 48.7(12.1)
2.0 67.8(18.6) 57.5(13.6) 47.9(9.8)
2.5 66.9(15.4) 59.1(14.2) 46.0(7.4)
Now Quessy et al. [12] gives an explicit formula for τ(˜
Ca1,a2) where Cis the Gaussian copula, but it is impossible
to get an expression for τ(˜
Ca1,a2) in the general case. However, one can use numerical integration to compute values
for both τ(˜
Ca1,a2) and ρS(˜
Ca1,a2). Figure 2displays the behavior of these two measures depending on the values of a1
and a2. One can see that for some given values of a2, the dependence is not necessarily monotonic in a1. In the case of
a2=0 and a1=, which correspond to the dependence between Z1and Z2
2, where Z1=Φ1(U1) and Z2=Φ1(U2),
one can notice that the values of Kendall’s tau and Spearman’s rho seem to be zero for the Gaussian, Student, and
Frank copulas. This is indeed true as shown next since these copulas are invariant with respect to the 180 rotation.
Proposition 1. Suppose that C180 =C. Then τ˜
C,0=ρS˜
C,0=0.
Proof. Since C180 =C, it follows that the law of (U1,U2) is the same as the law of (1U1,1U2), so (Z1,Z2) has the
7
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
Kendall's tau
Gaussian
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
0.6
Spearman's rho
Gaussian
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
Kendall's tau
Student
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
0.6
Spearman's rho
Student
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
Kendall's tau
Clayton
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
0.6
Spearman's rho
Clayton
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
Kendall's tau
Frank
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
0.6
Spearman's rho
Frank
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
Kendall's tau
Gumbel
0 0.5 1 1.5 2 2.5 3
a1
-0.2
0
0.2
0.4
0.6
Spearman's rho
Gumbel
a2=0
a2=0.5
a2=2.5
a1=a2
Figure 2: Graph of Kendall’s tau and Spearman’s rho for ˜
Cawhen the copula Cis Gaussian, Student, Clayton, Frank, and Gumbel with Kendall’s
tau 0.5, as a function of a1[0,3], for a2=0, a2=0.5, a2=2.5, and a2=a1.
same law as (Z1,Z2). Since ˜
C,0is the distribution function of (U1,|2U21|), it follows that E{(1 U1)|2U21|} =
E{U1|2U21|}, yielding that E(U1|2U21|)=1/4, so ρS˜
C,0=0 from formula (7). Finally, it follows from
(6) that if ( ˜
Z1,˜
Z2) is an independent copy of (Z1,Z2), PZ1˜
Z1,Z2
2˜
Z2
2=PZ1≤ −˜
Z1,Z2
2˜
Z2
2yielding that
PZ1˜
Z1,Z2
2˜
Z2
2=1/4. As a result, one gets from (5) that τ˜
C,0=0.
5. Tail behavior
The tail behavior of copulas is important for predicting joint extreme events occurrences and it is used for risk
management in many fields; it is for example a good proxy for systemic risk. In the bivariate case, recall that if
(U,V)D, for a given copula D, the lower tail dependence coecient λL(D) and the upper tail dependence coecient
8
λU(D) are defined respectively, if the limits exist, by λL(D)=lim
u0P(Vu|Uu)=lim
u0
D(u,u)
uand
λU(D)=lim
u1P(V>u|U>u)=lim
u1
12u+D(u,u)
1u
=λL(D180).
In this section, the tail behavior of non-central squared copulas is investigated. Quessy and Durocher [11] solved
the general case when a1=a2=0, while Quessy et al. [12] formulated a conjecture for the chi-square copula with
a1=a2>0. Here, the complete results for the tail behavior of ˜
Ca1,a2are stated in Theorem 2, whose proof is given in
Appendix B. Finally, some examples of computations are given.
Theorem 2. If the copula C has a continuous density c on (0,1)2, then λL(˜
Ca1,a2)=0for any a1,a20. If λU(C)
exists, then λU(˜
Ca1,a2)=λU(C)for any a1,a2>0. If a1=a2=0, then
λU(˜
C0,0)=1
2{λL(C)+λL(C90)+λU(C)+λL(C270)},
provided these coecients exist. Finally, if a2>0, then λU(˜
C0,a2)=lim
u0
C180(u/2,u)
u
+lim
u0
C270(u/2,u)
u, if these limits
exist, while if a1>0, then λU(˜
Ca1,0)=lim
u0
C90(u,u/2)
u
+lim
u0
C180(u,u/2)
u,if these limits exist.
Remark 4. Note that for any bivariate copula Dwith lower tail index λL(D),
1
2λL(D)lim inf
u0
D(u,u/2)
ulim sup
u0
D(u,u/2)
uλL(D).
In particular, if λL(D)=0, then lim
u0
D(u,u/2)
u
=lim
u0
D(u/2,u)
u
=0. For example, if Dis the Clayton copula with
parameter α > 0, then or any b(0,1], lim
u0
D(bu,u)
u
=lim
u0
D(u,bu)
u
=1+bα1.
Example 3 (Gaussian copula).For the Gaussian copula with parameter ρ(1,1), using to Theorem 2and Remark
4,λL(˜
Ca1,a2)=λU(˜
Ca1,a2)=0 for a1,a20. The same results hold for the Frank and Plackett copulas [9, Examples
5.21, 5.22], or any copula Cwith zero lower and upper tail indexes.
Example 4 (Clayton copula).As in Quessy and Durocher [11], one gets λL(C90)=λL(C180)=λL(C270 )=0 and
λL(C)=21. As a result, according to Theorem 2and Remark 4,λU(˜
C0,0)=1
2λL(C)=211, and λU(˜
Ca1,a2)=0
otherwise.
Example 5 (Gumbel copula).As in Quessy and Durocher [11], λU(C)=λL(C180)=221, and λL(C)=λL(C90)=
λL(C270)=0, for any α > 1. Theorem 2entails that λU(˜
C0,0)=1211and λU(˜
Ca1,a2)=221whenever
min(a1,a2)>0. However, for any a>0, λU˜
C0,a=λU˜
Ca,0=3
2(1+2α)1, since lim
u0
C180(bu,u)
u
=1+b
(1+bα)1for any b(0,1].
Example 6 (Student copula).Let Fνbe the cdf of a Student univariate distribution with ν > 0 degrees of freedom.
Then, as x→ ∞,xν¯
Fν(x) converges to νν/21
B(1/2,ν/2) , where ¯
Fν(x)=1Fν(x) and Bis the Beta function. Now suppose that
9
b>0 and let h(x,y)=1
2π1ρ21+x22ρxy+y2
ν(1ρ2)(ν+2)/2
be the density of a bivariate Student distribution with correlation
coecient ρ(1,1) and νdegrees of freedom. Further let Cρ,ν be the associated copula. It is well known that the
90 and 270 degrees rotations of Cρ,ν is Cρ,ν, while the 180 degrees rotation is the same as the original copula. It then
follows that if a(x)=F1
νn1b¯
Fν(x)o, then using the change of variable s=y/x,t=z/x, one gets
lim
u0
Cρ,ν(bu,u)
u
=lim
x→∞ R
a(x)R
xh(y,z)dydz
¯
F(x)
=lim
x→∞ x2R
a(x)
xR
1h(xs,xt)d sdt
¯
F(x).
Now, as x→ ∞,a(x)/xb1 , so
lim
x→∞ x2R
a(x)
xR
1h(xs,xt)d sdt
¯
F(x)
=B 1
2,ν
2!ν2(1 ρ2)(ν+1)/2
2πZ
b1Z
1s22ρst +t21ν/2dsdt
=2¯
Fν+1
s(ν+1)(1 ρ)
1+ρ
,(8)
where (8) is from Demarta and McNeil [3]. In particular, the limits in Theorem 2exist for b=1 and b=1/2,
λL(Cρ,ν)=λU(Cρ,ν)=B(1/2, ν/2) ν2(1 ρ2)(ν+1)/2
2πZ
1Z
1s22ρst +t21ν/2dsdt,
so λU(˜
Cρ,ν,0,0)=λU(Cρ,ν)+λU(Cρ,ν) and λU(˜
Cρ,ν,a1,a2)=λU(Cρ,ν) whenever min(a1,a2)>0. Setting λ(2)
ρ=
B(1/2, ν/2) ν2(1ρ2)(ν+1)/2
2πR
21R
1s22ρst +t21ν/2dsdt, one gets that λU(˜
Cρ,ν,a,0)=λU(˜
Cρ,ν,0,a)=λ(2)
ρ,ν +λ(2)
ρ,ν for
any a>0.
6. Conclusion
In this paper, new families of multivariate copulas depending on non-centrality parameters were introduced, ex-
tending the chi-square copulas, the Fisher copula, and squared copulas. The results show that by varying the non-
centrality parameters, one can model non-monotonic dependence. This is illustrated in the behavior of dependence
measures such as Kendall’s tau and Spearman’s rho. In addition, one finds the limiting behavior of the copula as some
non-centrality parameters tend to infinity, which has consequences for the estimation. Finally, the tail behavior of
these copulas is investigated. As a result, the conjecture of Quessy et al. [12] is solved.
7. Acknowledgments
The author is grateful to the Editor Yimin Xiao and two anonymous referees for their comments and suggestions.
This research is partially supported by the Canadian Statistical Sciences Institute (CANSSI) and the Fonds qu´
eb´
ecois
de la recherche sur la nature et les technologies (FRQNT).
10
Appendix A. Auxiliary results
Proposition 2. For any a>0 and any u(0,1), aG1
a(u)<0, a˜
ha(u)<0, and a˜
ha(u)<0.
Proof. Since aGa(x)={φ(xa)φ(x+a)}=2φ(x)ea2/2sinh(ax)<0 for any a,x>0, it follows that aG1
a(u)>
0 for any a>0 and u(0,1). As a result, a˜
ha(u)=φnaG1
a(u)on1aG1
a(u)o<0. Finally, a˜
ha(u)=
a˜
ha(u), since ˜
ha(u)=u+˜
ha(u).
Proposition 3. For any a0,
lim
u1
˜
ha(u)
1u
=
1/2,a=0;
0,a>0.
and lim
u1
1˜
ha(u)
1u
=
1/2,a=0;
1,a>0.
(A.1)
Proof. Recall that ˜
ha(u)=u+˜
ha(u). The result for a=0 is trivial since ˜
ha(u)=1+u
2. It suces to prove (A.1) when
a>0, the proof for ˜
h(a) being similar. First, note that
lim
u1
˜
ha(u)
1u
=lim
x→∞
Φ(ax)
1Φ(xa)+Φ(ax)=lim
x→∞
Φ(x)
1Φ(x2a)+Φ(x).(A.2)
Now, for any x>0, it is known that x
x2+1φ(x)Φ(x)φ(x)/x. Then, as x→ ∞,1Φ(x2a)
Φ(x)
(x2a)
1+(x2a)2e(x2a)2/2
1
xex2/2
=x(x2a)
(x2a)2+1e2a2+2ax → ∞. It then follows from (A.2) that lim
u1
˜
ha(u)
1u
=0.
Proposition 4. Suppose that α(u)/uaand β(u)/ubas u0. Then for any bivariate copula D,
lim
u0
D{α(u), β(u)}
uD(au,bu)
u
=0.(A.3)
Proof. For the Lipschitz property of copulas, as u0,
D{α(u)(u)}
uD(au,bu)
u
α(u)
ua+
β(u)
ub0.
Appendix B. Proofs of the theorems
Appendix B.1. Proof of Theorem 1
Proof. Recall that for any v[0,1] and any b0, ˜
hb(v)˜
hb(v)=vand ˜
hb(v)Φ(b). For j∈ {1,...,d}, set Aj=
(0,uj], ˜
Aj=˜
haj(uj),˜
haj(uj)i. If U=(U1,...,Ud)C., then ˜
Ca(u)=Pd
j=1{Uj˜
Aj}and ˜
C,...,,ak+1,...,ad(u)=
Pk
j=1{UjAj} ∩d
j=k+1{Uj˜
Aj}.As a result,
˜
Ca(u)˜
C,...,,ak+1,...,ad(u)max nPk
j=1{Uj˜
Aj}\∩k
j=1{UjAj},Pk
j=1{UjAj}\∩k
j=1{Uj˜
Aj}o
max
k
X
j=1
PUj˜
Aj\Aj,
k
X
j=1
PUjjAj\˜
Aj
kΦ(b),
11
since
PUj˜
Aj\AjPUj(uj,˜
haj(uj)haj(uj)Φ(aj)Φ(b)
and
PUjAj\˜
AjPUj(0,˜
haj(uj)=haj(uj)Φ(aj)Φ(b).
Appendix B.2. Proof of Theorem 2
Proof. For u>0,
˜
Ca1,a2(u,u)=Z˜
ha1(u)
˜
ha1(u)Z˜
ha2(u)
˜
ha2(u)
c(s,t)dsdt =u2c{Φ(a1), Φ(a2)}+Z˜
ha1(u)
˜
ha1(u)Z˜
ha2(u)
˜
ha2(u)
[c(s,t)c{Φ(a1), Φ(a2)}]dsdt.
Since cis continuous on (0,1)2, and both ˜
ha(u) and ˜
ha(u) converge to Φ(a)(0,1) as u0, it follows that
˜
Ca1,a2(u,u)/u2c{Φ(a1), Φ(a2)}as u0. Hence λL(˜
Ca1,a2)=0. For any copula D,λU(D)=2limu11˜
D(u,u)
1uif
the limit exists. Now
1˜
Ca1,a2(u,u)=PnU1˜
ha1(u)o+PnU1>˜
ha1(u)o+PnU2˜
ha2(u)o+PnU2>˜
ha2(u)o
PnU1˜
ha1(u),U2˜
ha2(u)oPnU1˜
ha1(u),U2>˜
ha2(u)o
PnU1>˜
ha1(u),U2˜
ha2(u)oPnU1>˜
ha1(u),U2>˜
ha2(u)o
=2(1 u)PnU1˜
ha1(u),U2˜
ha2(u)oPnU1˜
ha1(u),U2>˜
ha2(u)o
PnU1>˜
ha1(u),U2˜
ha2(u)oPnU1>˜
ha1(u),U2>˜
ha2(u)o.
As a result,
λU˜
Ca1,a2=lim
u1
PnU1˜
ha1(u),U2˜
ha2(u)o
1u
+lim
u1
PnU1˜
ha1(u),U2>˜
ha2(u)o
1u
+lim
u1
PnU1>˜
ha1(u),U2˜
ha2(u)o
1u
+lim
u1
PnU1>˜
ha1(u),U2>˜
ha2(u)o
1u,
provided these limits exist. Without loss of generality, assume a1a2. The case a1=a2=0 have been treated in
Quessy and Durocher [11] and follows readily from the last equality. So assume that a2>0, the case a1>0 being
similar. It then follows from (A.1) that limu1
P{U1˜
ha1(u),U2˜
ha2(u)}
1u=limu1
P{U1>˜
ha1(u),U2˜
ha2(u)}
1u=0. Furthermore, if
a1>0, then lim
u1
PnU1˜
ha1(u),U2>˜
ha2(u)o
1u
=0. Next, using Proposition 2, one gets from that
PnU1>˜
ha2(u),U2>˜
ha2(u)oPnU1>˜
ha1(u),U2>˜
ha2(u)oPnU1>˜
ha1(u),U2>˜
ha1(u)o.
12
As a result, using (A.1), it follows that for any a>0,
lim
u1
PnU1>˜
ha(u),U2>˜
ha(u)o
1u
=lim
u1
PnU1>˜
ha(u),U2>˜
ha(u)o
1˜
ha(u)
1˜
ha(u)
1u
=λU(C).
Hence, if a1,a2>0, then lim
u1
PnU1>˜
ha1(u),U2>˜
ha2(u)o
1u
=λU(C), proving that λU˜
Ca1,a2=λU(C) whenever
a1,a2>0. Finally, if a1=0, then ˜
h0(u)=(1 u)/2, and (A.1) and (A.3) yield
lim
u1
PnU1˜
ha1(u),U2>˜
ha2(u)o
1u
=lim
u0
PnU1u/2,1U2<1˜
ha2(1 u)o
u
=lim
u0
C270(u/2,u)
u,
if the latter exists. Similarly,
lim
u1
PnU1>˜
h0(u),U2>˜
ha2(u)o
1u
=lim
u0
Pn1U1u/2,1U2<1˜
ha2(1 u)o
u
=lim
u0
C180(u/2,u)
u,
if the latter exists. This completes the proof.
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