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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(V≤v|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=(X−0.5)2+, with X∼U(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=Φ−1(U1) and Z2=Φ−1(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 (|2U1−1|,|2U2−1|).

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=Φ−1(U1),...,Zd=Φ−1(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 (Zj−aj)2=(−Zj+aj)2, so −Zj=Φ(1 −Uj), leading to multivariate rotations of the original copula C. In

fact, in the bivariate case, ˜

C−a1,a2=˜

C90,a1,a2,˜

C−a1,−a2=˜

C180,a1,a2, and ˜

Ca1,−a2=˜

C270,a1,a2, where the rotations of Care

C90(u1,u2)=u2−C(1 −u1,u2), which is the copula of (1 −U1,U2), C180(u1,u2)=u1+u2−1+C(1 −u1,1−u2),

which is the copula of (1 −U1,1−U2), and C270(u1,u2)=u1−C(u1,1−u2), which is the copula of (U1,1−U2). 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)G−1

a(|u|)−a, and Ga(x)=P{|Z+a| ≤ x}=Φ(x−a)−Φ(−x−a),x≥0. It follows that the

non-central squared copula ˜

Cais given by

˜

Ca(u1,...,ud)=Ph∩d

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)=P∩k

j=1{Uj≤uj} ∩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 k≤dparameters are large, the general case being obtained as a permutation of the indices.

Theorem 1. Set a=(a1,...,ad)∈[0,∞)dand suppose that a j≥b for all j ∈ {1,...,k}, with 1≤k≤d. 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 aj≥b, 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

|F−1

1(U1)+a1|, . . . |F−1

1(U1)+a1|is the copula of V1,...,Vd, where Vj=|Uj−Fj{−2aj−F−1

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 aj≥bj.

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 aﬀect 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 coeﬃcients 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{U1≤V1,U2≤V2}.(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=Φ−1(U1) and Z2=Φ−1(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 (1−U1,1−U2), 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,|2U2−1|), it follows that E{(1 −U1)|2U2−1|} =

E{U1|2U2−1|}, yielding that E(U1|2U2−1|)=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=P−Z1≤ −˜

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 coeﬃcient λL(D) and the upper tail dependence coeﬃcient

8

λU(D) are deﬁned respectively, if the limits exist, by λL(D)=lim

u↓0P(V≤u|U≤u)=lim

u↓0

D(u,u)

uand

λU(D)=lim

u↑1P(V>u|U>u)=lim

u↑1

1−2u+D(u,u)

1−u

=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,a2≥0. 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 coeﬃcients exist. Finally, if a2>0, then λU(˜

C0,a2)=lim

u↓0

C180(u/2,u)

u

+lim

u↓0

C270(u/2,u)

u, if these limits

exist, while if a1>0, then λU(˜

Ca1,0)=lim

u↓0

C90(u,u/2)

u

+lim

u↓0

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

u↓0

D(u,u/2)

u≤lim sup

u↓0

D(u,u/2)

u≤λL(D).

In particular, if λL(D)=0, then lim

u↓0

D(u,u/2)

u

=lim

u↓0

D(u/2,u)

u

=0. For example, if Dis the Clayton copula with

parameter α > 0, then or any b∈(0,1], lim

u↓0

D(bu,u)

u

=lim

u↓0

D(u,bu)

u

=1+b−α−1/α.

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,a2≥0. 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)=2−1/α. As a result, according to Theorem 2and Remark 4,λU(˜

C0,0)=1

2λL(C)=2−1−1/α, and λU(˜

Ca1,a2)=0

otherwise.

Example 5 (Gumbel copula).As in Quessy and Durocher [11], λU(C)=λL(C180)=2−21/α, and λL(C)=λL(C90)=

λL(C270)=0, for any α > 1. Theorem 2entails that λU(˜

C0,0)=1−21/α−1and λU(˜

Ca1,a2)=2−21/α whenever

min(a1,a2)>0. However, for any a>0, λU˜

C0,a=λU˜

Ca,0=3

2−(1+2−α)1/α, since lim

u↓0

C180(bu,u)

u

=1+b−

(1+bα)1/α for 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/2−1

B(1/2,ν/2) , where ¯

Fν(x)=1−Fν(x) and Bis the Beta function. Now suppose that

9

b>0 and let h(x,y)=1

2π√1−ρ21+x2−2ρxy+y2

ν(1−ρ2)−(ν+2)/2

be the density of a bivariate Student distribution with correlation

coeﬃcient 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 C−ρ,ν, while the 180 degrees rotation is the same as the original copula. It then

follows that if a(x)=F−1

νn1−b¯

Fν(x)o, then using the change of variable s=y/x,t=z/x, one gets

lim

u↓0

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)/x→b−1/ν , 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∞

b−1/ν Z∞

1s2−2ρst +t2−1−ν/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∞

1s2−2ρst +t2−1−ν/2dsdt,

so λU(˜

Cρ,ν,0,0)=λU(Cρ,ν)+λU(C−ρ,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(1−ρ2)(ν+1)/2

2πR∞

21/ν R∞

1s2−2ρst +t2−1−ν/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), ∂aG−1

a(u)<0, ∂a˜

ha(−u)<0, and ∂a˜

ha(u)<0.

Proof. Since ∂aGa(x)=−{φ(x−a)−φ(x+a)}=2φ(x)e−a2/2sinh(ax)<0 for any a,x>0, it follows that ∂aG−1

a(u)>

0 for any a>0 and u∈(0,1). As a result, ∂a˜

ha(−u)=φn−a−G−1

a(u)on−1−∂aG−1

a(u)o<0. Finally, ∂a˜

ha(u)=

∂a˜

ha(−u), since ˜

ha(u)=u+˜

ha(−u).

Proposition 3. For any a≥0,

lim

u↑1

˜

ha(−u)

1−u

=

1/2,a=0;

0,a>0.

and lim

u↑1

1−˜

ha(u)

1−u

=

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 suﬃces to prove (A.1) when

a>0, the proof for ˜

h(a) being similar. First, note that

lim

u↑1

˜

ha(−u)

1−u

=lim

x→∞

Φ(−a−x)

1−Φ(x−a)+Φ(−a−x)=lim

x→∞

Φ(−x)

1−Φ(x−2a)+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→ ∞,1−Φ(x−2a)

Φ(−x)≥

(x−2a)

1+(x−2a)2e−(x−2a)2/2

1

xe−x2/2

=x(x−2a)

(x−2a)2+1e−2a2+2ax → ∞. It then follows from (A.2) that lim

u↑1

˜

ha(−u)

1−u

=0.

Proposition 4. Suppose that α(u)/u→aand β(u)/u→bas u→0. Then for any bivariate copula D,

lim

u↓0

D{α(u), β(u)}

u−D(au,bu)

u

=0.(A.3)

Proof. For the Lipschitz property of copulas, as u→0,

D{α(u),β(u)}

u−D(au,bu)

u≤

α(u)

u−a+

β(u)

u−b→0.

Appendix B. Proofs of the theorems

Appendix B.1. Proof of Theorem 1

Proof. Recall that for any v∈[0,1] and any b≥0, ˜

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)=P∩d

j=1{Uj∈˜

Aj}and ˜

C∞,...,∞,ak+1,...,ad(u)=

P∩k

j=1{Uj∈Aj} ∩d

j=k+1{Uj∈˜

Aj}.As a result,

˜

Ca(u)−˜

C∞,...,∞,ak+1,...,ad(u)≤max nP∩k

j=1{Uj∈˜

Aj}\∩k

j=1{Uj∈Aj},P∩k

j=1{Uj∈Aj}\∩k

j=1{Uj∈˜

Aj}o

≤max

k

X

j=1

PUj∈˜

Aj\Aj,

k

X

j=1

PUj∈jAj\˜

Aj

≤kΦ(−b),

11

since

PUj∈˜

Aj\Aj≤PUj∈(uj,˜

haj(uj)≤haj(−uj)≤Φ(−aj)≤Φ(−b)

and

PUj∈Aj\˜

Aj≤PUj∈(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 u→0, it follows that

˜

Ca1,a2(u,u)/u2→c{Φ(−a1), Φ(−a2)}as u↓0. Hence λL(˜

Ca1,a2)=0. For any copula D,λU(D)=2−limu↑11−˜

D(u,u)

1−uif

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)o−PnU1≤˜

ha1(−u),U2>˜

ha2(u)o

−PnU1>˜

ha1(u),U2≤˜

ha2(−u)o−PnU1>˜

ha1(u),U2>˜

ha2(u)o

=2(1 −u)−PnU1≤˜

ha1(−u),U2≤˜

ha2(−u)o−PnU1≤˜

ha1(−u),U2>˜

ha2(u)o

−PnU1>˜

ha1(u),U2≤˜

ha2(−u)o−PnU1>˜

ha1(u),U2>˜

ha2(u)o.

As a result,

λU˜

Ca1,a2=lim

u↑1

PnU1≤˜

ha1(−u),U2≤˜

ha2(−u)o

1−u

+lim

u↑1

PnU1≤˜

ha1(−u),U2>˜

ha2(u)o

1−u

+lim

u↑1

PnU1>˜

ha1(u),U2≤˜

ha2(−u)o

1−u

+lim

u↑1

PnU1>˜

ha1(u),U2>˜

ha2(u)o

1−u,

provided these limits exist. Without loss of generality, assume a1≤a2. 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 limu↑1

P{U1≤˜

ha1(−u),U2≤˜

ha2(−u)}

1−u=limu↑1

P{U1>˜

ha1(u),U2≤˜

ha2(−u)}

1−u=0. Furthermore, if

a1>0, then lim

u↑1

PnU1≤˜

ha1(−u),U2>˜

ha2(−u)o

1−u

=0. Next, using Proposition 2, one gets from that

PnU1>˜

ha2(u),U2>˜

ha2(u)o≤PnU1>˜

ha1(u),U2>˜

ha2(u)o≤PnU1>˜

ha1(u),U2>˜

ha1(u)o.

12

As a result, using (A.1), it follows that for any a>0,

lim

u↑1

PnU1>˜

ha(u),U2>˜

ha(u)o

1−u

=lim

u↑1

PnU1>˜

ha(u),U2>˜

ha(u)o

1−˜

ha(u)

1−˜

ha(u)

1−u

=λU(C).

Hence, if a1,a2>0, then lim

u↑1

PnU1>˜

ha1(u),U2>˜

ha2(u)o

1−u

=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

u↑1

PnU1≤˜

ha1(−u),U2>˜

ha2(u)o

1−u

=lim

u↓0

PnU1≤u/2,1−U2<1−˜

ha2(1 −u)o

u

=lim

u↓0

C270(u/2,u)

u,

if the latter exists. Similarly,

lim

u↑1

PnU1>˜

h0(u),U2>˜

ha2(u)o

1−u

=lim

u↓0

Pn1−U1≤u/2,1−U2<1−˜

ha2(1 −u)o

u

=lim

u↓0

C180(u/2,u)

u,

if the latter exists. This completes the proof.

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