perturbation of bivariate copulas

JID:FSS AID:6533 /FLA [m3SC+; v 1.191; Prn:8/05/2014; 16:37] P.1 (1-14)

Available online at www.sciencedirect.com

ScienceDirect

Fuzzy Sets and Systems ••• (••••) •••–•••www.elsevier.com/locate/fss

Perturbation of bivariate copulas

Radko Mesiar a, Magda Komorníková a,∗, Jozef Komorník b

a Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakiab Faculty of Management, Comenius University, Odbojárov 10, P.O. BOX 95, 820 05 Bratislava, Slovakia

Received 3 December 2013; received in revised form 7 March 2014; accepted 21 April 2014

Abstract

New types of constructions of bivariate copulas are introduced, discussed and exemplified. Based on a given copula C, we lookfor its perturbation into another copula CH , possibly close to C. A special stress is put on the perturbation of the basic copulas M ,W , Π . Our results generalize several methods known from the literature, such as the Farlie–Gumbel–Morgenstern copula family,for example. An illustrative example when fitting copulas to real data is also added.© 2014 Elsevier B.V. All rights reserved.

Keywords: Farlie–Gumbel–Morgenstern copulas; Perturbation of product copula; Radially symmetric copula; Construction method for copula

1. Introduction

Copulas, especially their bivariate form, have obtained a growing interest in the last two decades, especially becauseof their numerous applications. Recall only that a function C : [0,1]2 → [0,1] is called a (bivariate) copula [24]whenever it is

i) 2-increasing, i.e.,

VC

([u1, u2] × [v1, v2]) = C(u1, v1) + C(u2, v2) − C(u1, v2) − C(u2, v1) ≥ 0

for all 0 ≤ u1 ≤ u2 ≤ 1, 0 ≤ v1 ≤ v2 ≤ 1 (recall that VC([u1, u2] × [v1, v2]) is the C-volume of the rectangle[u1, u2] × [v1, v2]);

ii) grounded, i.e., C(u,0) = C(0, v) = 0 for all u,v ∈ [0,1];iii) it has a neutral element e = 1, i.e., C(u,1) = u and C(1, v) = v for all u,v ∈ [0,1].

For more details, examples and applications we recommend monographs Joe (1997) [13] and Nelsen (2006) [22].Fitting of an appropriate copula to real data is one of major tasks in application of copulas. For this purpose, a large

* Corresponding author.E-mail addresses: [email protected] (R. Mesiar), [email protected] (M. Komorníková),

[email protected] (J. Komorník).

http://dx.doi.org/10.1016/j.fss.2014.04.0160165-0114/© 2014 Elsevier B.V. All rights reserved.

http://www.sciencedirect.com

http://dx.doi.org/10.1016/j.fss.2014.04.016

http://www.elsevier.com/locate/fss

mailto:[email protected]





2 R. Mesiar et al. / Fuzzy Sets and Systems ••• (••••) •••–•••

buffer of potential copulas is necessary, preferably parametric families of copulas. Once we know approximately acopula C appropriate to model the observed data, we look for a minor perturbation of C which fit better than C itself.This is, e.g., the case of Farlie–Gumbel–Morgenstern (FGM) class of copulas, all of them being a perturbation of theindependence copula Π , Π(u,v) = uv. Recall that FGM family (CFGM

α )α∈[−1,1] of copulas is given by

CFGMα (u, v) = uv + αu(1 − u)v(1 − v) (1)

see [8,11,21].Several generalizations of FGM approach to perturb the product copula Π can be found in literature, see, for

example, [2,3,9,12,18,23].As another example, consider the comonotonicity copula M , M(u,v) = min(u, v), and a parametric noise family

Hα : [0,1]2 → [0,1], α ∈ [0,1] given by Hα(u, v) = α(max(u, v) − 1). Then, for α ∈ ]0,1[,MHα(u, v) = max

(0,M(u, v) + Hα(u, v)

) = max(0, (1 − α)M(u, v) + α(u + v − 1)

)defines a singular copula with support on 3 segments connecting the point ( α

1+α, α

1+α) with vertices (0,1), (1,1) and

(1,0) (if α = 0, MH0 = M ; if α = 1, MH1 = W is the countermonotonicity copula given by W(u,v) = max(0, u +v − 1)).

Obviously, also some other copulas perturbations have attracted several researchers, adding some noise to a givencopula C, compare [1,14]. We have presented one such approach at AGOP 2013 conference [19], and the aim of thispaper is a deeper discussion of a general approach covering all mentioned perturbation methods.

For a given copula C : [0,1]2 → [0,1], we will look for constraints on the noise H : [0,1]2 → � so that thefunction CH : [0,1]2 → [0,1] given by

CH (u, v) = max(0,C(u, v) + H(u,v)

)(2)

is also a copula. Obviously FGM copulas given by (1) are linked to C = Π and Hα(u, v) = αu(1 − u)v(1 − v)

(observe that in this case, no truncation is necessary).Recall that for each copula C it holds W ≤ C ≤ M , and thus the formula (2) can be considered also in the form

CH (u, v) = max(W(u,v),min

(M(u,v),C(u, v) + H(u,v)

)).

Our aim was to generalize the approach exemplified in the two above examples, and thus we work in this paper withformula (2).

The paper is organized as follows. The next section is devoted to several examples and discusses the relationshipbetween C and H . In Section 3, we discuss perturbations of the product copula Π . Section 4 deals with some pertur-bation of the copula M (and related perturbations of the copula W ). In Section 5, we discuss perturbations of radiallysymmetric copulas. Section 6 brings an illustrative example of fitting copulas to real data. Finally, some concludingremarks are added.

2. Perturbation of bivariate copulas

For a fixed copula C : [0,1]2 → [0,1], consider the function CH given by (2). To satisfy the groundedness conditionof copulas by CH , necessarily H(u,0) ≤ 0 and H(0, v) ≤ 0 for all u,v ∈ [0,1]. Similarly, e = 1 is a neutral elementof CH only if H(u,1) = H(1, v) = 0 for all u,v ∈ [0,1]. The main problem to ensure that CH is a copula is toguarantee the 2-increasingness of CH , which depends both on C and H .

Example 1. Consider the Hamacher product copula CH (it belongs also to Clayton copulas, Ali–Mikhail–Haq copu-las, etc.) given by

CH (u, v) = uv

u + v − uv

whenever (u, v) �= (0,0). For α ∈ �, let Hα : [0,1]2 → [0,1] be given by

Hα(u, v) = α(u2 − u3)(v2 − v3).


R. Mesiar et al. / Fuzzy Sets and Systems ••• (••••) •••–••• 3

Evidently, Hα(u,0) = Hα(0, v) = Hα(u,1) = Hα(1, v) = 0 for all u,v ∈ [0,1], and thus CHHα

given by (2) is a copula

if and only if it is 2-increasing. After some processing with software MATHEMATICA it can be shown that CHHα

is acopula if and only if α ∈ [−2,1].

In general, if C is a singular copula, the function H �= 0 cannot be absolutely continuous. Similarly, if C is anabsolutely continuous copula, H cannot be singular. Therefore, as a special case of the perturbation (2), one can dealwith perturbation related to functions f,g : [0,1] → [0,1] and constant λ ∈ � in the form already discussed in [19],namely

Cλ,f,g(u, v) = max(0,C(u, v) + λC

(f (u), f (v)

)). (3)

Obviously, FGM family given in (1) can be seen as a special case of construction (3), considering C = Π , λ ∈ [−1,1]and f = g given by f (x) = x − x2. Note that as a necessary condition to ensure that e = 1 is a neutral element ofCλ,f,g , one should consider f (1) = g(1) = 0. On the other hand, if λ ≤ 0, then Cλ,f,g is always grounded. However, ifλ > 0, then one should consider C(f (0), g(0)) = 0 (which is trivially satisfied for any copula C if f (0) = g(0) = 0).

As a particular case of construction (3), recall the next result from [23].

Proposition 1. Let f,g : [0,1] → [0,1] be Lipschitz continuous functions with Lipschitz constants c and d , respec-tively, and let f (0) = g(0) = f (1) = g(1) = 0. Then the function Πλ,f,g : [0,1]2 → [0,1] given by

Πλ,f,g(u, v) = max(0, uv + λf (u)g(v)

)is a copula whenever |λcd| ≤ 1.

Observe that the bounds for the constant λ in Proposition 1 need not be the best ones, see Example 3. In fact, theconstraints on λ are related to the range of f ′(u)g′(v), u,v ∈ [0,1] (in points where the considered derivatives exist),which is always a subset of [−cd, cd].

Another special instance of construction (3), fixing λ = −1, was studied in [1].

Proposition 2. Let N : [0,1] → [0,1] be an involutive decreasing bijection (i.e., a strong negation) such that it is1-Lipschitz continuous on the interval [k,1], where k is the fixed point of N , N(k) = k. Then the function M−1,N,N :[0,1]2 → [0,1] given by

M−1,N,N (u, v) = max(0,min(u, v) − min

(N(u),N(v)

))is a copula.

In the case when no truncation is necessary, we have two general results.

Proposition 3. Let C : [0,1]2 → [0,1] be a copula and H : [0,1]2 → [0,1] be a function so that C + H ≥ 0 and CH

is a copula, i.e., CH = C + H is a copula. Then also CλH = C + λH is a copula for each λ ∈ [0,1].

Proof. It is enough to realize that CλH = C + λH = (1 − λ)C + λCH , i.e., CλH is a convex combination of copulasC and CH . �

Observe that Proposition 3 may fail if the truncation in formula (2) applies. As an example, consider a strongnegation N given by N(x) = max(1−9x, 1−x

9 ). Putting H(u,v) = −min(N(u),N(v)), MH is a copula due to Propo-sition 2. However, MH/2(

111 , 1

11 ) = 0 and MH/2(111 , 1

10 ) = 9220 > 2

220 = 110 − 1

11 , violating the 1-Lipschitz continuityof the function MH/2, and thus MH/2 is not a copula.

Proposition 4. Under the constraints of Proposition 3, the function CH is a copula, where C : [0,1]2 → [0,1] is thesurvival copula related to C,

C(u, v) = u + v − 1 + C(1 − u,1 − v),



Fig. 1. Volumes of 9 rectangles.

and H : [0,1]2 → [0,1] is given by

H (u, v) = H(1 − u,1 − v).

Proof. It is enough to observe that CH = CH . �Corollary 1. Let C be radially symmetric (C and its survival copula C coincide) and let H : [0,1]2 → � be a function.Then the following are equivalent:

i) C + H is a copula;ii) C + H is a copula.

Example 2. Continuing in Example 1, recall that the survival copula CH : [0,1]2 → [0,1] is given by

CH (u, v) = uv(2 − u − v)

1 − uv

whenever (u, v) �= (1,1). Due to Proposition 4, for any α ∈ [−2,1], the function CH

Hα: [0,1]2 → [0,1] given by (for

(u, v) �= (1,1))

CH

Hα(u, v) = uv(2 − u − v)

1 − uv+ αuv(1 − u)2(1 − v)2

is a copula.

We have a next perturbation method valid for any copula C.

Theorem 1. Let C : [0,1]2 → [0,1] be a copula and define HCλ : [0,1]2 → [0,1], λ ∈ [0,1] by

HCλ (u, v) = λ

(u − C(u, v)

)(v − C(u, v)

).

Then CHCλ

: [0,1]2 → [0,1] given by

CHCλ(u, v) = C(u, v) + λ

(u − C(u, v)

)(v − C(u, v)

)(4)

is a copula for each λ ∈ [0,1] and any copula C.

Proof. The boundary conditions are trivially satisfied, and thus we have to show the 2-increasingness of CHCλ

only.Consider a fixed rectangle [u1, u2] × [v1, v2] and denote the C-volumes of 9 rectangles as depicted in Fig. 1 bya1, · · · , a9. Observe that a1 + a4 + a7 = u1, a7 + a8 + a9 = v1, etc. Obviously, VC([u1, u2] × [v1, v2]) = a5. For thefunction C C it holds
Hλ



VCHC

λ

([u1, u2] × [v1, v2])

= a5 + λ((a1 + a2)(a6 + a9) + (a1 + a4)(a8 + a9) − a1(a5 + a6 + a8 + a9) − a9(a1 + a2 + a4 + a5)

)= a5 + λ(a2a6 + a4a8 − a1a5 − a5a9) = λ(a2a6 + a4a8) + a5

(1 − λ(a1 + a9)

) ≥ 0,

proving the 2-increasingness of CHCλ

. Thus CHCλ

is a copula, independently of C and λ ∈ [0,1]. �Perturbation methods (4) proposed in Theorem 1 can be consecutively repeated. So, for example, CHW

1= Π , CHΠ

λ

is FGM copula with parameter λ, etc.

Remark 1. There is an interesting relation between the perturbation method (4) and M-based ordinal sums. Indeed,consider C = (〈ak, bk,Ck〉 | k ∈K)M , i.e.

C(u, v) ={

ak + (bk − ak)Ck(u−ak

bk−ak,

v−ak

bk−ak) if (u, v) ∈ ]ak, bk[2 for some k ∈K,

min(u, v) otherwise.

Then

CHCλ

=(⟨

ak, bk,CH

Ckλ(bk−ak)

⟩ ∣∣∣ k ∈ K)

M.

So, for example, if C = (〈0, 12 ,W 〉, 〈 1

2 ,1,W 〉)M , then CHC1

= (⟨0, 1

2 , Π+W2

⟩,⟨ 1

2 ,1, Π+W2

⟩)M

, because of CHW0.5

=Π+W

2 .

3. Perturbations of the product copula Π

We have already recalled in Proposition 1 due to [23] a family of functions H such that ΠH is a copula, generalizingthe approach used when constructing FGM family. Observe that each polynomial p is Lipschitz continuous on [0,1],and considering the constraints p(0) = p(1) = 0, clearly p : [0,1] → � is given by p(x) = x(1 − x)q(x), whereq is some polynomial (possibly a constant, as a polynomial of degree 0). Hence for any polynomials q , h there isa non-zero constant λ so that for Hλ : [0,1]2 → � given by

Hλ(u, v) = λu(1 − u)v(1 − v)q(u)h(v)

it holds ΠHλ = Π + Hλ, and ΠHλ is a copula. Obviously, the set of all such constants λ is a closed interval [α,β]such that α < 0 < β .

Example 3. Let f,g : [0,1] → � be given by f (x) = x(1−x)(1+x) and g(y) = y(1−y)y. Then f ′(x) = 1−3x2 ∈[−2,1] and g′(y) = 2y − 3y2 ∈ [−1, 1

3 ]. Therefore, f ′(x)g′(y) ∈ [−1,2], and due to results of [23], the functionΠHλ : [0,1]2 → � generated by Hλ : [0,1]2 → �, Hλ(u, v) = λf (u)g(v), is a copula if and only if λ ∈ [− 1

2 ,1].Due to Corollary 1, also the function ΠHλ

: [0,1]2 → �,

ΠHλ(u, v) = uv + λf (1 − u)g(1 − v) = uv + λu(1 − u)(2 − u)v(1 − v)2

is a copula if and only if λ ∈ [− 12 ,1].

For any decreasing function f : [0,1] → �, put H(u,v) = f (u)f (v) and denote ΠH by Πf . If f (1) = 0, it isevident that the function Πf : [0,1]2 → [0,1] given by

Πf (u, v) = max(0, uv − f (u)f (v)

)is a semicopula [4,6], i.e., Πf is grounded, e = 1 is its neutral element, and Πf is non-decreasing in each coordinate.To check whether Πf is a copula we have to check the 2-increasingness only.

Theorem 2. Let N : [0,1] → [0,1] be a convex strong negation. Then the function ΠN : [0,1]2 → [0,1] given by

ΠN(u,v) = max(0, uv − N(u)N(v)

)(5)

is a negative quadrant dependent copula.



Proof. Observe first that due to the convexity of N , the one-side derivatives N ′(x+) and N ′(x−) exist in each pointx ∈ ]0,1[ (and also N ′(1−) and N ′(0+) exist). More, due to the involutivity of N , N = N−1 and thus if xy =N(x)N(y), necessarily N(x) = y, and N ′(x−)N ′(y+) = N ′(x+)N ′(y−) = 1. Obviously, ΠN given by (5) satisfiesΠN(u,v) = 0 if and only if uv ≤ N(u)N(v), i.e., v ≤ N(u).

As already mentioned, ΠN is a semicopula, and thus it is enough to show its 2-increasingness on its positive area(with boundary line being the graph of N ). Consider 0 < u1 < u2 ≤ 1, 0 < v1 < v2 ≤ 1 such that u1v1 ≥ N(u1)N(v1).Then the volume VΠN

of the rectangle [u1, u2] × [v1, v2] is non-negative if and only if

(u2 − u1)(v2 − v1) ≥ (N(u1) − N(u2)

)(N(v1) − N(v2)

).

From the convexity and decreasingness of N , it follows(N(u1) − N(u2)

)(N(v1) − N(v2)

) ≤ (u1 − u2)N′(u+

1

)(v1 − v2)N

′(v+1

)≤ (u2 − u1)(v2 − v1)N

′(u+1

)N ′(v−

1

) = (u2 − u1)(v2 − v1),

proving the desired fact that ΠN is a copula. Evidently, ΠN ≤ Π , and thus ΠN is negative quadrant dependent. �Observe that due to [25], each strong negation N is generated by an automorphism ϕ : [0,1] → [0,1], N(x) =

ϕ−1(1 − ϕ(x)). It is not difficult to check that if ϕ is concave then N is convex. So, for example, for any p ∈ (0,1],ϕp : [0,1] → [0,1] given by ϕp(x) = xp is a concave automorphism, and the corresponding strong negation Np :[0,1] → [0,1] given by Np(x) = (1 − xp)

1p is convex.

Then the function ΠNp : [0,1]2 → [0,1] given by

ΠNp(u, v) = max(0, uv − ((

1 − up)(

1 − vp)) 1

p)

is a negative quadrant dependent copula, with boundary member ΠN1 = W (observe that N1 given by N1(x) = 1 − x

is the standard strong negation, N1 = Ns , also called Zadeh negation).

Example 4. For c ∈ ]0,1[, define a function Nc : [0,1] → [0,1] by

Nc(x) ={

1 − 1−cc

x if x ∈ [0, c]c(1−x)

1−celse.

Then Nc is a strong negation which is convex if and only if c ∈ ]0, 12 ].

Applying formula (5), we see that ΠNc : [0,1]2 → [0,1] is given by

ΠNc(u, v) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

c2u+c2v−c2+(1−2c)u·v(1−c)2 if (u, v) ∈ [c,1]2,

(1−c)u+(v−1)c1−c

if u ∈ [0, c] and v ≥ c−(1−c)uc

,(1−c)v+(u−1)c

1−cif v ∈ [0, c] and u ≥ c−(1−c)v

c,

0 else.

Observe that for each c ∈ ]0,1[, ΠNc is a semicopula which is Lipschitz continuous with constant max(1, ( c1−c

)2),

i.e., for c > 12 , ΠNc is not a copula. On the other hand, for each c ∈ ]0, 1

2 ], ΠNc is a copula.

Theorem 2 can be strengthen in the next way.

Corollary 2. Let N : [0,1] → [0,1] be a convex strong negation, and let λ ∈ ]0,1]. Define fλ : [0,1] → [0,1] by

fλ(x) = λN

(min

(1,

x

λ

)).

Then Πfλ : [0,1]2 → [0,1] given by

Πfλ(u, v) = max

(0, uv − λ2N

(min

(1,

u

λ

))N

(min

(1,

v

λ

)))is a negative quadrant dependent copula.



Note that in several cases also gλ = λN can be considered, λ ∈ ]0,1], N a convex strong negation. However, toensure that Πgλ is a copula, we need to ensure that if xy = λ2N(x)N(y), then λ2N ′(x)N ′(y) ≤ 1 in all points x, y

where the derivative N ′(x),N ′(y) exist.

Example 5. Consider the standard negation given by Ns(x) = 1 − x. Observe that for any u,v ∈ ]0,1[, N ′s(u) =

N ′s(v) = −1, and thus 1 ≥ λ2N ′

s(u)N ′s(v) = λ2 for any λ ∈ ]0,1]. Then

i) Πgλ : [0,1]2 → [0,1] given by

Πgλ(u, v) = max(0, uv − λ2(1 − u)(1 − v)

) = max(0,

(1 − λ2)uv + λ2(u + v − 1)

)is a Sugeno–Weber copula [15] for any λ ∈ ]0,1];

ii) Πfλ : [0,1]2 → [0,1] given by

Πfλ(u, v) ={

max(0, λu + λv − λ2) if (u, v) ∈ [0, λ]2,

uv else

is a copula for each λ ∈ ]0,1]. Observe that Πfλ has a singular part with mass λ2 uniformly distributed overthe segment connecting points (0, λ) and (λ,0), and its absolutely continuous part has density 1 on the domain[0,1]2 \ [0, λ]2. Observe that each copula Πfλ can be seen as a Π -ordinal sum, Πfλ = (〈0, λ,W 〉)Π , recentlyintroduced in [17].

4. Perturbations of copula M

We have already recalled a result from [1], see Proposition 2, showing that the function M−1,N,N : [0,1]2 → [0,1]given by

M−1,N,N (u, v) = max(0,min(u, v) − min

(N(u),N(v)

))is a copula for any strong negation N which is 1-Lipschitz continuous on the interval [k,1], k being the fixed pointof N . Observe that each convex strong negation satisfies these constraints. Proposition 2 can be generalized as follows.

Theorem 3. Let f : [0,1] → [0,1] be a non-increasing function such that f (1) = 0, f (k) = k for some k ∈ ]0,1[,and f is 1-Lipschitz continuous on the interval [k,1]. Then the function M−1,f,f : [0,1]2 → [0,1] given by

M−1,f,f (u, v) = max(0,min(u, v) − min

(f (u), f (v)

))is a copula.

Proof. It is not difficult to check that the function M−1,f,f has 3 formulae on 3 subsets forming the partition of thedomain [0,1]2:

1. M−1,f,f (u, v) = 0 if v ≤ min(u,f (u)) or u ≤ min(v, f (v));2. M−1,f,f (u, v) = v − f (u) if f (u) < v ≤ u;3. M−1,f,f (u, v) = u − f (v) otherwise.

Evidently, M−1,f,f is a quasi-copula, i.e., 1-Lipschitz semicopula. To see the 2-increasingness of M−1,f,f , observefirst that volumes of rectangles completely contained in the closure of any of 3 above mentioned subsets of [0,1]2 arevanishing.

Due to the additivity of volumes, it is enough to check the rectangles with opposite vertices belonging to theboundaries of discussed subsets of [0,1]2, so that these rectangles have one part in one subset, and the remaining partin another subset of the domain we are taking into consideration. The non-negativity of their volumes in the case thattwo opposite vertices belong to the zero area of M−1,f,f (thus also third vertex is necessarily in that zero area) followsfrom the monotonicity of M−1,f,f .

Finally, let k ≤ u < v and consider the square [u,v]2. Then

VM−1,f,f

([u,v]2) = v − f (v) + u − f (u) − 2(u − f (v)

) = v − u − (f (u) − f (v)

) ≥ 0

due to 1-Lipschitzianity of f on [k,1]. �



As an example we recall copulas MHα introduced in Section 1, which can be written in the form MHα = M−1,fα,fα ,α ∈ [0,1], where fα(u) = α(1 − u).

We introduce another example.

Example 6. For λ ∈ [0,1], let gλ : [0,1] → [0,1] be given by gλ(x) = max(λ − x,0). Then M−1,gλ,gλ is a singularcopula given by

M−1,gλ,gλ(u, v) = max(0,min(u, v) − min

(max(λ − u,0),max(λ − v,0)

)).

Observe that M−1,gλ,gλ is the M-based ordinal sum, see [22],

M−1,gλ,gλ = (〈0, λ,W 〉)M

.

Another class of functions f yielding a copula M−1,f,f is given in the next theorem. Before starting the theo-rem, recall first that a function f : [0,1] → [0,1] is called unimodal, with modal value m ∈ [0,1], if f/[0,m] isnon-decreasing and f/[m,1] is non-increasing.

Theorem 4. Let f : [0,1] → [0,1] be a unimodal 1-Lipschitz continuous function such that f (0) = f (1) = 0. Thenf (x) ≤ x for all x ∈ [0,1] and the function M−1,f,f : [0,1]2 → [0,1] given by

M−1,f,f (u, v) = min(u, v) − min(f (u), f (v)

)is a singular copula.

Proof. Let m be a modal point argument, i.e., f is non-decreasing on [0,m], and it is non-increasing on [m,1]. Ifm ∈ {0,1} then f (x) = 0 for all x ∈ [0,1], and thus M−1,f,f = M . Suppose m ∈ ]0,1[. From the unimodality andcontinuity of f , together with boundary constraints f (0) = f (1) = 0, it follows that there is a decreasing bijectionϕ : [0,1] → [0,1] such that ϕ(m) = m, and such that f (x) = f (ϕ(x)) for each x ∈ [0,1]. Then

M−1,f,f (u, v) =

⎧⎪⎨⎪⎩

u − f (u) if u ≤ v ≤ ϕ(u),

v − f (v) if v ≤ u ≤ ϕ−1(v),

v − f (u) if ϕ(u) ≤ v ≤ u,

u − f (v) otherwise.

It is obvious that M−1,f,f is a quasi-copula, and its 2-increasingness can be shown in a similar way as in the case ofTheorem 3. �

Observe that if a function f satisfies the constraints of Theorem 4, then also λf satisfies these constraints for eachλ ∈ [0,1]. Moreover, each opposite diagonal section ω of a copula C, ω(x) = C(x,1 − x), is 1-Lipschitz continuousand satisfies ω(0) = ω(1). Thus, if ω is unimodal, M−1,ω,ω is a copula. This copula has a diagonal section δ given byδ(x) = M−1,ω,ω(x, x) = x − ω(x), and M−1,ω,ω is the smallest copula with diagonal section δ, i.e., M−1,ω,ω = Bδ isthe Bertino copula [22]. For related considerations, see also [7]. The next corollary brings a complete description offunctions f such that M−1,f,f is a copula.

Corollary 3. Let f : [0,1] → [0,1] be a function. Then M−1,f,f : [0,1]2 → [0,1] given by

M−1,f,f = max(0,min(u, v) − min

(f (u), f (v)

))is a copula if and only if the function f : [0,1] → [0,1] given by

f (x) = min(x,f (x)

)satisfies the constraints of Theorem 4, i.e., f is 1-Lipschitz continuous unimodal function and f (1) = f (1) = 0.

Proof. The sufficiency follows from Theorem 4 (observe that f (0) = 0 for any f : [0,1] → [0,1]). To see the neces-sity, observe first that M−1,f,f = M−1,f ,f

, and its diagonal section δ is given by δ(x) = x − f (x). Supposing M−1,f,f

is a copula, necessarily δ(1) = 1, and thus f (1) = f (1) = 0. Moreover, δ is 2-Lipschitz continuous, increasing and



Fig. 2. Formulae for the copula M1,f,f from Example 7.

δ ≤ id[0,1], and hence f is 1-Lipschitz. Suppose that f is not unimodal, i.e., there are x, y, z ∈ [0,1], x < y < z, suchthat x ≥ f (x) > f (y) < f (z) = f (x). Then M−1,f,f (x, y) = x − f (y) > M−1,f,f (x, z) = x − f (x), violating theincreasing monotonicity of M−1,f,f . Hence necessarily f is unimodal. �Example 7.

i) For any Archimedean copula C generated by an additive generator h : [0,1] → [0,∞] (h is convex, strictlydecreasing and h(1) = 0, see [22]),

C(u, v) = h−1(min(h(0), h(u) + h(v)

)),

its opposite diagonal section f : [0,1] → [0,1], f (x) = h−1(min(h(0), h(x)+h(1−x))), satisfies the constraintsof Theorem 4 with modal argument m = 1

2 , and ϕ(x) = 1 − x. Consider, e.g., the product copula Π . Thenf (x) = Π(x,1 − x) = x(1 − x), and for each λ ∈ [0,1],

Mλ,f,f (u, v) =

⎧⎪⎨⎪⎩

(1 − λ)u + λu2 if u ≤ v ≤ 1 − u,

(1 − λ)v + λv2 if v ≤ u ≤ 1 − v,

v + λ(u2 − u) if 1 − u ≤ v ≤ u,

u + λ(v2 − v) otherwise,

determines a singular copula.The copula M1,f,f is depicted in Fig. 2.

ii) Consider the opposite diagonal section of a W -ordinal sum copula C = (〈0, 12 ,M〉, 〈 1

2 ,1,M〉)W , see [20], givenby

f (x) = C(x,1 − x) ={

min(x, 12 − x) if x ∈ [0, 1

2 ],min(x − 1

2 ,1 − x) otherwise.

Then f is 1-Lipschitz continuous, f (0) = f (1) = 0, but f is not unimodal (two modal arguments are m1 = 14

and m2 = 34 ).

Then the section M−1,f,f (x, 34 ) = 2x − 1 if x ∈ [ 1

4 , 12 ], i.e., M−1,f,f is not 1-Lipschitz continuous and thus not a

copula.

Remark 2. Observe that each function f : [0,1] → [0,1] satisfying the constraints of Theorem 4 which is symmetricwrt point 0.5, i.e., f (x) = f (1 − x) for all x ∈ [0,1], can be seen as an opposite diagonal section of some symmetriccopula. Then also M−1,f,f is a symmetric copula (with support on the main and opposite diagonals) and its oppositediagonal section g : [0,1] → [0,1] is given by

g(x) = M−1,f,f (x,1 − x) = min(x − f (x),1 − x − f (1 − x)

).

Thus also M−1,g,g is a symmetric copula and

M−1,g,g(x,1 − x) = f (x)



for all x ∈ [0,1]. Thus the repeated application of Theorem 4 can be seen as a method of constructing a symmetriccopula with a given opposite diagonal section. So, for example, consider f (x) = x(1 − x), the opposite diagonalsection of the product copula Π . Then, compare also Fig. 2, g(x) = min(x2, (1 − x)2), and

M−1,g,g(u, v) =

⎧⎪⎨⎪⎩

u − u2 if u ≤ v ≤ 1 − u,

v − v2 if v ≤ u ≤ 1 − v,

v − (1 − u)2 if 1 − u ≤ v ≤ u,

u − (1 − v)2 otherwise.

5. Perturbations of radially symmetric copulas

As already shown, applying formula (3) and considering the standard negation Ns , Ns(x) = 1 − x, it holds

M−1,Ns ,Ns = Π−1,Ns ,Ns = W−1,Ns ,Ns = W.

A natural question arises: when C−1,Ns ,Ns is a copula?

Theorem 5. Let C : [0,1]2 → [0,1] be a copula. Then the following are equivalent:

i) C is radially symmetric (C and its survival copula C coincide), i.e.,

C(u, v) = C(u, v) = u + v − 1 + C(1 − u,1 − v)

(see [16,22]);ii) C−1,Ns ,Ns is a copula;

iii) C−1,Ns ,Ns = W .

Proof. (iii) ⇒ (ii) is trivial. Suppose C−1,Ns ,Ns is a copula. Observe that for each copula C it holds C ≥ W , i.e.,C(u,1 − u) ≥ 0. Suppose C(u,1 − u) − C(1 − u,u) > 0. Then C(1 − u,u) − C(u,1 − u) < 0 and due to thecontinuity of copulas, there is ε > 0 such that C(1−u,u+ ε)−C(u+ ε,1−u) < 0, i.e., C−1,Ns ,Ns (1−u,u+ ε) = 0.This contradicts the fact that C−1,Ns ,Ns (1 − u,u + ε) ≥ W(1 − u,u + ε) = ε > 0. Therefore, C−1,Ns ,Ns (u,1 − u) = 0for all u ∈ [0,1], i.e., C−1,Ns ,Ns = W , proving (ii) ⇒ (iii). Moreover, from C−1,Ns ,Ns = W it follows that C(x, y) −C(1 − x,1 − y) = x + y − 1 whenever x + y ≥ 1, i.e.,

C(x, y) = x + y − 1 + C(1 − x,1 − y) = C(x, y).

On the other hand, again for x + y ≥ 1, put 1 − x = u,1 − y = v. Then u + v ≤ 1, and C(u, v) = C(1 − x,1 − y) =C(x, y) − x − y + 1 = C(1 − u,1 − v) + u + v − 1 = C(u, v). Summarizing, C = C, i.e., (ii) ⇒ (i).

Suppose C = C. Then

C−1,Ns ,Ns (u, v) = max(0,C(u, v) − C(1 − u,1 − v)

)= max

(0, C(u, v) − C(1 − u,1 − v)

) = max(0, u + v − 1) = W(u,v),

i.e., (i) ⇒ (iii). Now, the proof is complete. �We have also the next consequence of Theorem 5.

Corollary 4. Let C be a radially symmetric copula and let λ ∈ [0,1]. Then the function C−λ,Ns,Ns : [0,1]2 → [0,1]given by

C−λ,Ns,Ns (u, v) = max(0,C(u, v) − λC(1 − u,1 − v)

)is a copula.

Proof. Observe that due the radially symmetry of C, it holds

C−λ,Ns,Ns (u, v) = max(0,C(u, v) − λ

(1 − u + 1 − v − 1 + C(u, v)

))= max

(0, (1 − λ)C(u, v) + λ(u + v − 1)

).



Fig. 3. Description of copula C−λ,Ns ,Ns from Example 8.

Fig. 4. Real estate investment trust indexes.

Evidently, C−λ,Ns,Ns is a quasi-copula, and its 2-increasingness follows from the 2-increasingness of the functionK : [0,1]2 → [0,1] given by

K(u,v) = (1 − λ)C(u, v) + λ(u + v − 1). �.

Example 8. Consider the ordinal sum copula C = (〈0, 12 ,W 〉, 〈 1

2 ,1,W 〉)M . Then C = C, and C−λ,Ns,Ns , λ ∈ [0,1], isa singular copula depicted in Fig. 3.

6. Application to real data modeling

We have investigated the relations between USA and Japan daily returns of the Real Estate Investment Trust (REIT)indexes (from 3.1.2000 to 8.5.2012) in different time periods, determined by the recent global financial markets crises(August 1, 2008–April 30, 2009) that can be also clearly identified from Fig. 4, presenting the parallel developmentof the considered REIT indexes.

We have performed filtering of the returns of two individual REIT indexes (in order to avoid a possible violationof the i.i.d. property) by ARMA–GARCH models (separately for the individual considered time subperiods). Forall three time subperiods and all couples of (filtered) returns of the REIT indexes we have performed the nonpara-metric correlation analyses (based on the Kendall coefficients). We have observed that the values of the correlationcoefficients have dropped substantially between the first and the second considered time subperiods and even moredramatically for the third subperiod (see Table 1).



Table 1The values of the Kendall’s correlation coefficient τ for the all time periods.

03.01.2000–31.07.2008 01.08.2008–30.04.2009 01.05.2009–08.05.2012

τ 0.73 0.27 0.06

Table 2The overview of optimal types of copulas for the filtered returns of REIT indexes for precrisis subperiod.

Type of copula λ θ AD

CGθ x 1.999 4.422

CGθ + λ(u − CG

θ )(v − CGθ ) 0.75 1.999 3.707

CClθ x 1.476 4.957

CClθ + λ(u − CCl

θ )(v − CClθ ) 1 1.476 3.156

CJθ x 2.248 12.086

CJθ + λ(u − CJ

θ )(v − CJθ ) 1 2.248 9.497

CFθ x 6.602 6.610

CFθ + λ(u − CF

θ )(v − CFθ ) 0.056 6.602 6.581

Table 3The overview of optimal types of copulas for the filtered returns of REIT indexes for crisis subperiod.


CGθ x 1.313 2.565

CGθ + λ(u − CG

θ )(v − CGθ ) 0.249 1.313 2.308

CClθ x 0.485 1.853

CClθ + λ ∗ (u − CCl

θ )(v − CClθ ) 0.493 0.485 1.332

CJθ x 1.397 4.093

CJθ + λ(u − CJ

θ )(v − CJθ ) 0.556 1.397 3.090

CFθ x 2.607 2.250

CFθ + λ(u − CF

θ )(v − CFθ ) 0 2.607 2.250

Subsequently we have applied the fitting by copulas to the residuals of ARMA–GARCH filters. We consideredmodels from Joe (CJ

θ ), Frank (CFθ ), strict Clayton (CCl

θ ) and Gumbel (CGθ ) families as well as their perturbations

given by (4). For estimation of parameters for each type of models we have used the maximum pseudo-likelihoodmethod described e.g. in [10]. The optimal models for all 3 time periods are presented in Tables 2, 3 and 4. For all ofthem, the simulation based GoF test yielded p-value > 0.1.

For selecting the optimal models we have applied the Kolmogorov–Smirnov–Anderson–Darling (KSAD, for whichwe have used the abbreviation AD) [5] test statistic (for which we also constructed a GoF simulation based test).

We see that for all 3 considered time periods the minimal value of AD was received for perturbed model of thetype (4) (derived from the Clayton type copula). Therefore, the perturbation proposed by (4), can provide applicablesolutions.

7. Concluding remarks

We have introduced several methods for perturbations of binary copulas, resulting to new construction methods forcopulas. Even if the perturbed copula is symmetric, the proposed methods can also be used for constructing parametricfamilies of non-symmetric copulas, see, e.g., Example 3. As an important topic for the further investigation we openthe problem of modification of copulas of higher dimensions. Some extensions of results recalled or introduced inthis paper are obvious. For example, considering the approach linked to FGM family it is not difficult to check that



Table 4The overview of optimal types of copulas for the filtered returns of REIT indexes for postcrisis subperiod.


CGθ x 1.048 8.806

CGθ + λ(u − CG

θ )(v − CGθ ) 0.086 1.048 8.546

CClθ x 0.173 2.256

CClθ + λ(u − CCl

θ )(v − CClθ ) 0 0.173 2.256

CJθ x 1.038 10.323

CJθ + λ(u − CJ

θ )(v − CJθ ) 0.186 1.038 9.499

CFθ x 0.564 9.186

CFθ + λ(u − CF

θ )(v − CFθ ) 0 0.564 9.186

for each 1-Lipschitz continuous functions fi : [0,1] → [0,1] such that fi(0) = fi(1) = 0, i = 1, · · · , n, the functionDλ : [0,1]n → [0,1] given by

Dλ(u1, · · · , un) =n∏

i=1

ui + λ

n∏i=1

fi(ui)

is an n-ary copula for each λ ∈ [−1,1].On the other side, we cannot directly generalize Theorem 4 or Corollary 3. Considering the function D : [0,1]n →

[0,1] given by

D(u1, · · · , un) = max(0,min(u1, · · · , un) − min(1 − u1, · · · ,1 − un)

),

it is not difficult to check that for each 2-dimensional marginal function Di,j with fixed uk = 1 whenever k /∈ {i, j} itholds

Di,j (ui, uj ) = max(0,min(ui, uj ) − 0

) = min(ui, uj ).

Supposing D is a copula, then Di,j = M for each i, j ∈ {1, · · · , n}, i �= j , if and only if D(u1, · · · , un) =min(u1, · · · , un) for all (u1, · · · , un) ∈ [0,1]n, what is a contradiction whenever n > 2.

Acknowledgements

We express our gratitude to the FSS editors and anonymous referees whose comments and suggestions have helpedus to improve the original version of this paper. This work was supported by Slovak Research and DevelopmentAgency under contracts No. APVV–0496–10, APVV–0236–12 and by VEGA 1/0143/11.

References

[1] I. Aguiló, J. Suner, J. Torrens, A construction method of semicopulas from fuzzy negations, Fuzzy Sets Syst. 226 (2013) 99–114.[2] C. Amblard, S. Girard, A new symmetric extension of FGM copulas, Metrika 70 (2009) 1–17.[3] I. Bairamov, S. Kotz, Dependence structure and symmetry of Huang–Kotz FGM distributions and their extensions, Metrika 56 (2002) 55–72.[4] B. Bassano, F. Spizzichino, Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes, J. Multivar. Anal.

93 (2) (2005) 313–339.[5] D. Berg, H. Bakken, Copula goodness-of-fit tests: a comparative study, Working paper, University of Oslo and Norwegian Computing Center,

2006.[6] F. Durante, C. Sempi, Semicopulae, Kybernetika 41 (3) (2005) 315–328.[7] F. Durante, J. Fernández-Sanchéz, R. Pappadá, Copulas, diagonals and tail dependence, Fuzzy Sets Syst. (2014), http://dx.doi.org/10.1016/

j.fss.2014.03.014, in press.[8] D.G.J. Farlie, The performance of some correlation coefficients for a general Bivariate distribution, Biometrika 47 (1960) 307–323.[9] M. Fischer, I. Klein, Constructing symmetric generalized FGM copulas by means of certain univariate distributions, Metrika 65 (2007)

243–260.[10] C. Genest, A.C. Favre, Everything you always wanted to know about copula modeling but were afraid to ask, J. Hydrol. Eng. 12 (2007)

347–368.

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4153543A32303132s1


http://refhub.elsevier.com/S0165-0114(14)00191-2/bib424B3A32303032s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib62617373616E6F3A32303035s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib62617373616E6F3A32303035s1



http://refhub.elsevier.com/S0165-0114(14)00191-2/bib647572616E74653A32303035s1


http://refhub.elsevier.com/S0165-0114(14)00191-2/bib6661726C69653A31393630s1



http://refhub.elsevier.com/S0165-0114(14)00191-2/bib474632303037s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib474632303037s1




[11] E.J. Gumbel, Bivariate exponential distributions, J. Am. Stat. Assoc. 55 (1960) 698–707.[12] J.S. Huang, S. Kotz, Modifications of the Farlie–Gumbel–Morgenstern distribution. A tough hill to climb, Metrika 49 (1999) 135–145.[13] H. Joe, Multivariate Models and Dependence Concepts, Chapman and Hall, London, 1997.[14] J.-M. Kim, E.A. Sungur, T. Choi, T.-Y. Heo, Generalized bivariate copulas and their properties, Model Assist. Stat. Appl. 6 (2011) 127–136.[15] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic – Studia Logica Library, vol. 8, Kluwer Academic Publishers, Dordrecht,

2000.[16] E.P. Klement, R. Mesiar, E. Pap, Invariant copulas, Kybernetika 38 (3) (2002) 275–286.[17] A. Kolesárová, R. Mesiar, J. Kalická, On a new construction of 1-Lipschitz aggregation functions, quasi-copulas and copulas, Fuzzy Sets Syst.

226 (2013) 19–31.[18] C.D. Lai, M. Xie, A new family of positive quadrant dependence bivariate distributions, Stat. Probab. Lett. 46 (2000) 359–364.[19] R. Mesiar, J. Komorník, M. Komorníková, On some construction methods for bivariate copulas, in: Aggregation Functions in Theory and in

Practise: Proceedings of the 7th International Summer School on Aggregation Operators at the Public University of Navarra, Pamplona, Spain,16.–20.6.2013, in: Advances in Intelligent Systems and Computing, vol. 228, 2013, pp. 39–46.

[20] R. Mesiar, J. Szolgay, W -ordinals sums of copulas and quasi-copulas, in: MAGIA & UWPM 2004: Proceedings of the Mathematical Seminar,Kocovce, 2004, STU v Bratislave SvF, ISBN 80-227-2165-4, 2004, pp. 78–83.

[21] D. Morgenstern, Einfache Beispiele Zweidimensionaler Verteilungen, Mitt.bl. Math. Stat. 8 (1956) 234–235.[22] R.B. Nelsen, An Introduction to Copulas, second edition, Springer Series in Statistics, Springer-Verlag, New York, 2006.[23] J.A. Rodríguez-Lallena, M. Úbeda-Flores, A new class of bivariate copulas, Stat. Probab. Lett. 66 (2004) 315–325.[24] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Stat. Univ. Paris 8 (1959) 229–231.[25] E. Trillas, Sobre funciones de negación en la teoría de conjuntos difusos, Stochastica III (1) (1979) 47–59 (in Spanish) (Reprinted (English

version), in: S. Barro, A. Sobrino, A. Bugarin (Eds.), Advances of Fuzzy Logic, Universidad de Santiago de Compostela, 1998, pp. 31–43.)

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib67756D62656C3A31393630s1


http://refhub.elsevier.com/S0165-0114(14)00191-2/bib6A6F653A31393937s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib6B696D3A32303131s1




http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4B4D4B2D32303133s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4B4D4B2D32303133s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4C583A32303030s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4D4B4B3A32303133s1



http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4D533A32303034s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib4D533A32303034s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib6D6F7267656E737465726E3A31393536s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib6E656C73656E3A32303036s1


http://refhub.elsevier.com/S0165-0114(14)00191-2/bib736B6C6172s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib7472696C6C6173s1

http://refhub.elsevier.com/S0165-0114(14)00191-2/bib7472696C6C6173s1

perturbation of bivariate copulas

Documents