asymmetric copulas

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New families of symmetric/asymmetric copulas Radko Mesiar a,b , Vadoud Najjari c,∗ a Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia b Institute for Research and Applications of Fuzzy Modeling, Division of University Ostrava, National Super computing Center IT4 Innovations,

30. dubna 22, Ostrava 701 03, Czech Republic c Department of Mathematics, Islamic Azad University, Maragheh branch, Maragheh, Iran

Received 23 May 2013; received in revised form 2 December 2013; accepted 23 December 2013

Abstract In 2004, Rodríguez-Lallena and Úbeda-Flores have introduced a class of bivariate copulas which generalizes some known families such as the Farlie–Gumbel–Morgenstern distributions. In 2006, Dolati and Úbeda-Flores presented multivariate generalizations of this class, also they investigated symmetry, dependence concepts and measuring the dependence among the components of each classes. In this paper, a new method of constructing binary copulas is introduced, extending the original study of Rodríguez-Lallena and Úbeda-Flores to new families of symmetric/asymmetric copulas. Several properties and parameters of newly introduced copulas are included. Among others, relationship of our construction method with several kinds of ordinal sums of copulas is clarified. © 2013 Elsevier B.V. All rights reserved. Keywords: Copulas; Dependence concepts; Measures of association; Tails

1. Introduction Copulas are mathematical objects that fully capture the dependence structure among random variables. Since their introduction they have gained a lot of popularity in several fields like finance, insurance and reliability theory, etc. Copulas are a way of studying scale-free measures of dependence and also are a tool to build families of bivariate distributions with given margins, hence copulas are of interest to statisticians [8,16]. A copula is a function C : [0, 1]2 → [0, 1] which satisfies: (a) for every u, v in [0, 1], C(u, 0) = 0 = C(0, v) and C(u, 1) = u and C(1, v) = v; (b) for every u1 , u2 , v1 , v2 in [0, 1] such that u1  u2 and v1  v2 , VC (R) = C(u2 , v2 ) − C(u2 , v1 ) − C(u1 , v2 ) + C(u1 , v1 )  0 (in other words, for all rectangles R = [u1 , u2 ] × [v1 , v2 ] whose vertices lie in [0, 1]2 , VC (R)  0). Copulas allow us to combine univariate distributions to obtain a joint distribution with a particular dependence structure, see the famous Sklar theorem [18]. As a result of Sklar’s theorem, copulas link joint distribution functions to their one-dimensional margins. In the literature we can see wide effort in construction of new copulas. Recall, for example, conic copulas [9], univariate conditioning method proposed in [11], UCS (univariate conditioning stable) copulas [4], several construction * Corresponding author.

E-mail addresses: [email protected] (R. Mesiar), [email protected] (V. Najjari). 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.12.015

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methods based on diagonal or horizontal (vertical) sections discussed in [5–7]. Rodríguez-Lallena and Úbeda-Flores [16] introduced a class of bivariate copulas of the form: Cλ (u, v) = uv + λf (u)g(v),

(u, v) ∈ [0, 1]2

(1.1)

where f and g are two non-zero absolutely continuous functions such that f (0) = f (1) = g(0) = g(1) = 0 and the admissible range of the parameter λ is [−1/ max(αγ , βδ), −1/ min(αδ, βγ )] where     α = inf f  (u): u ∈ A < 0, β = sup f  (u): u ∈ A > 0     δ = sup g  (u): u ∈ B > 0 γ = inf g  (v): v ∈ B < 0,     B = v ∈ [0, 1]: g  (u) exists . (1.2) A = u ∈ [0, 1]: f  (u) exists , This class of copulas provides a method for constructing bivariate distributions with a variety of dependence structures and generalizes some known families such as the Farlie–Gumble–Morgenstern (FGM) distributions as well as the bivariate distributions introduced by Sarmanov in [17]. Dolati and Úbeda-Flores [3] provided procedures to construct parametric families of multivariate distributions which generalize (1.1). On the other hand, they have presented multivariate generalizations of this class. Also they studied the symmetry and dependence concepts, measuring the dependence among the components of each classes and provided several examples. Kim et al. [10] generalized the method of Rodríguez-Lallena and Úbeda-Flores to any given copula. They presented a construction, considering an arbitrary given copula C, as below: Cλ∗ (u, v) = C(u, v) + λf (u)g(v),

(u, v) ∈ [0, 1]2 .

(1.3)

Here for any rectangle R = [u1 , u2 ] × [v1 , v2 ], λ should satisfy −VC (R) −VC (R) λ  × max(αγ , βδ)  × min(αγ , βδ)

(1.4)

where α, β, γ , δ are the same as in (1.2),  = (u2 − u1 )(v2 − v1 ), u1 , u2 , v1 , v2 are in [0, 1] such that u1  u2 and v1  v2 and f , g are two non-zero absolutely continuous functions defined on [0, 1] such that f (0) = f (1) = g(0) = g(1) = 0. The method of Kim et al. gives only a poor sufficient condition for λ determination, it excludes positive λ’s and also it is in general rather difficult to be applied. Mesiar et al. [12] proposed a rather general construction method for bivariate copulas, generalizing some construction methods known from the literature. They have written Eq. (1.1) and Eq. (1.3) as below,     Cλ∗ (u, v) = max 0, C(u, v) − λΠ f (u), g(v) . (1.5) In this study we suggest how to generalize the study of Rodríguez-Lallena and Úbeda-Flores to new families of symmetric/asymmetric copulas. The extended new families have n  1 parameters and so they are more flexible than the method of Rodríguez-Lallena and Úbeda-Flores. Thus they are able to model the more miscellaneous structures of dependency. Several selected dependence measures such as Kendall’s tau, Spearman’s rho, Gini’s gamma and also the tails behaviors of these new families will be shown and at last several examples will be provided. The rest of this paper is structured as follows. In Section 2 we introduce some new copulas families as a modification of the product copula. Section 3 discusses some properties of the new families. In Section 4 several examples will be investigated. Relationships to some other construction methods are given in Section 5. Section 6 summarizes the conclusion of our work. 2. Extensions of product copula Let fi (u) and gi (v) for i = 1, 2, . . . , n be absolutely continuous real functions defined on [0, 1]. Then we consider the function given by C(u, v) = uv +

n  i=1

λi fi (u)gi (v),

n  1,

(2.1)

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for all u, v in [0, 1] and λi ∈ R. Our purpose is to determine the cases in which C in (2.1) is a copula (note that then C is an absolutely continuous copula). Of course, our model proposed by (2.1) is a generalized form of the model of Rodríguez-Lallena and Úbeda-Flores and for n = 1 we get their result. For C to satisfy boundary conditions (a), it is enough that fi (0) = fi (1) = gi (0) = gi (1) = 0 for i = 1, 2, . . . , n. Then the main problem in (2.1) is in restriction of the parameters λi , such that C is 2-increasing, namely, for all rectangles R = [u1 , u2 ] × [v1 , v2 ] whose vertices lie in [0, 1]2 , we have to examine, VC (R)  0.

(2.2)

By using Rodríguez-Lallena and Úbeda-Flores results from [16] and for i = 1, 2, . . . , n defining αi , βi , γi , δi by (i) means of (1.2), considering functions fi , gi it is evident that the functions Cτi (u, v) : [0, 1]2 → [0, 1] given by Cτ(i) (u, v) = uv + τi fi (u)gi (v) i

(2.3)

are copulas for any τi ∈ [−1/ max(αi γi , βi δi ), −1/ min(αi δi , βi γi )]. Then, for any a1 , . . . , an ∈ [0, 1] such that n n (i) 2 i=1 ai = 1, also the convex sum C = i=1 ai Cτi is a copula. It is not difficult to check that for all (u, v) ∈ [0, 1] it holds, C(u, v) = uv +

n 

ai τi fi (u)gi (v).

(2.4)

i=1

Denoting λi = ai τi , i = 1, . . . , n, we see that the n-tuples (λ1 , . . . , λn ) = (a1 τ1 , . . . , an τn ) partially solves our problem of characterization parameters λ1 , . . . , λn such that (2.1) yields a copula. We summarize the above observation in the next proposition. Proposition 2.1. Under thenotation introduced above, for each n-tuple (λ1 , . . . , λn ) = (a1 τ1 , . . . , an τn ) where a1 , . . . , an ∈ [0, 1] such that ni=1 ai = 1, and −1 −1  τi  max(αi γi , βi δi ) min(αi δi , βi γi )

(2.5)

for any i = 1, . . . , n, the function C(u, v) : [0, 1]2 → [0, 1] given by (2.1) is a copula. Proposition 2.1 gives only a sufficient condition for n-tuples (λ1 , . . . , λn ) of parameters bringing functions constructed by means of (2.1) to be copulas. Obviously investigating density function c of C in (2.1) is a way how to determine the n-tuples of parameters λi such that C is 2-increasing. The density function c : [0, 1]2 → R of the function C is given (almost everywhere) by c(u, v) = 1 +

n 

λi fi (u)gi (v).

(2.6)

i=1

Theorem 2.2. A function C : [0, 1]2 → [0, 1] given by (2.1) is a copula if and only if c(u, v) given by (2.6) is nonnegative in each point (u, v) where all derivatives fi (u) and gi (v), i = 1, 2, . . . , n, exist. As in general, finding the domains of parameters λi (for i = 1, 2, . . . , n) such that C is a copula may be complicated, we try to add a new sufficient method to solve our problem. Moreover, in Section 4 we will introduce several examples. Proposition 2.3. Using the notation from Proposition 2.1, an n-tuple (λ1 , . . . , λn ) makes a function C : [0, 1]2 → [0, 1] given by Eq. (2.1) a copula whenever the next inequalities are satisfied: n 

  λi di max(αi γi , βi δi ) + ei min(αi δi , βi γi )  −1

i=1

for all (d1 , . . . , dn ), (e1 , . . . , en ) ∈ {0, 1}n such that (d1 , . . . , dn ).(e1 , . . . , en ) = (d1 e1 , . . . , dn en ) = (0, . . . , 0).

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Proof. For arbitrary (u, v) ∈ [0, 1]2 such that all derivatives fi (u), gi (v), i = 1, . . . , n, exist, if fi (u)gi (v)  0, then 0  fi (u)gi (v)  max(αi γi , βi δi ). If λi  0 we put di = 0 = ei and then   λi fi (u)gi (v)  0 = λi di max(αi γi , βi δi ) + ei min(αi δi , βi γi ) . If λi < 0 we put di = 1, ei = 0 and then   λi fi (u)gi (v)  λi di max(αi γi , βi δi ) + ei min(αi δi , βi γi ) . Similarly we will proceed in the case when fi (u)gi (v)  0, to ensure the validity of inequality   λi fi (u)gi (v)  λi di max(αi γi , βi δi ) + ei min(αi δi , βi γi ) . Then for our choice of (d1 , . . . , dn ) and (e1 , . . . , en ) the following inequality holds, n  i=1

λi fi (u)gi (v) 

n 

  λi di max(αi γi , βi δi ) + ei min(αi δi , βi γi )  −1.

i=1

The last inequalities mean c(u, v)  0 almost everywhere if c(u, v) exists and thus C is 2-increasing. Hence C is a copula. 2 (1)

(1)

(k)

(k)

It is evident that for any n-tuples L1 = (λ1 , . . . , λn ), . . . , Lk = (λ1 , . . . , λn ) such that the corresponding func tion C1 , . . . , Ck given by (2.1) are copulas, also each convex combination L = ki=1 ai Li = (λ1 , . . . , λn ) yields a k copula C = i=1 ai Ci . Therefore, the set L ⊆ Rn of all solutions (λ1 , . . . , λn ) characterized in Theorem 2.2 is a convex set. Moreover, unexpectedly it is possible to show that both Proposition 2.1 and Proposition 2.3 characterize the same convex set L (we will exemplify this fact also by several examples in Section 4). We prove this fact for the case n = 2 only. To simplify the notation, denote for i = 1, 2: ai = max(αi γi , βi δi ) > 0,

bi = min(αi δi , βi γi ) < 0.

Then Proposition 2.3 characterize L by means of the next 8 inequalities: (1) (2) (3) (4) (5) (6) (7) (8)

λ1 a1  −1 λ1 b1  −1 λ 2 a2  λ 2 b2  λ 1 a 1 + λ2 a 2  λ 1 a 1 + λ2 b 2 

−1 −1 −1 −1

λ1 b1 + λ2 a2  −1 λ1 b1 + λ2 b2  −1.

(2.7) 8

Obviously (λ1 , λ2 ) = (0, 0) solves all inequalities and L = i=1 Li where Li is the solution of inequality (i). Inequality (5) determines the half-plane L5 containing the point (0, 0) and bounded by the straight line connecting −1 −1 −1 −1 points ( −1 a1 , 0) and (0, a2 ). Similarly L6 is related to points ( a1 , 0) and (0, b2 ), L7 is related to points ( b1 , 0) and −1 −1 −1 (0, −1 a2 ) and L8 is related to points ( b1 , 0) and (0, b2 ). Consequently, the set L is a convex closure of 4 points ( a1 , 0),

−1 −1 ( −1 b1 , 0), (0, a2 ) and (0, b2 ). However the same relation is obtained when analyzing Proposition 2.1. The extended new families have n  1 parameters and so they are more flexible than the original family, i.e., they are able to model the more miscellaneous structures of dependency.

3. Properties of the new class of copulas In this section we recall several selected dependence measures such as Kendall’s tau, Spearman’s rho, Gini’s gamma and also the tails behaviors applied on families in the model (2.1). Of course these measures have very important role in the application of copulas. Note that there are many copulas that have not closed form for these measures. In this

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section we see that the new proposed families via (2.1) have a closed form in the mentioned dependence measures and also in the tail behavior. The mentioned dependence measures are given for any copula C as below (for details see [15]): 1 1 τC = 1 − 4

∂C ∂C du dv ∂u ∂v

0 0

1 1 ρC = 12

C(u, v) du dv − 3 0 0

1 γC = 4



 C(u, u) + C(u, 1 − u) du − 2

0

C(t, t) t 1 − C(t, t) = 2 − lim . − 1−t t→1

tLC = lim

t→0+

tUC

(3.1)

Calculations of these measures for any family given by (2.1) are summarized in the following theorem whose proof is a matter of calculation only and therefore omitted. Theorem 3.1. Let C be a copula in Eq. (2.1). Then measures of association, Kendall’s tau, Spearman’s rho, Gini’s gamma and also the tail behavior of C, are respectively given by 1 1  n n  τC = −4 λi fi (u)gi (v) + u λi fi (u)gi (v) v

+

0 0 n 

i=1



λi fi (u)gi (v) ×

i=1

ρC = 12

n 

1 λi

1  n 0

tLC = lim

t→0+

tUC = lim

t→1−

i=1



λi fi (u)gi (v)

du dv

i=1

i=1

γC = 4

n 

1 fi (u) du

0

gi (v) dv 0

λi fi (u)gi (u) +

i=1

n

n 

 λi fi (u)gi (1 − u) du

i=1

i=1 λi fi (t)gi (t)

=0 t i=1 λi fi (t)gi (t) = 0. 1−t

n

(3.2)

The mentioned measures for product copula are zero, but as we can see in Theorem 3.1, new proposed families’ measures are related with functions fi , gi (for i = 1, 2, . . . , n) and hence can be different of zero up to the tail dependences. The fact that in these new families also tails dependency will be zero, follows from the absolute continuity and boundary conditions of functions fi , gi (for i = 1, 2, . . . , n). Indeed, if the derivatives of fi or gi are unbounded in a neighborhood of the point 0 (1), then the corresponding parameter λi is necessarily 0. On the other hand, supposing the boundedness (i.e., Lipschitz continuity) of both fi or gi in some neighborhood of the point 0 (1), then also the fractions fi (t)/t or gi (t)/t are bounded in that neighborhood, and then the result follows from the continuity and boundary conditions of fi and gi , fi (0) = gi (0) = fi (1) = gi (1) = 0.

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Fig. 1. Possible area for (λ1 , λ2 ) based on Propositions 2.1, 2.3 (Left), and based on Theorem 2.2 (Right).

4. Examples of the modified product copulas In this section the main aim is to describe examples of the proposed extensions of product copula. For convenience we will consider the case n = 2. Example 4.1. Let functions f1 , g1 , f2 , g2 : [0, 1] → R be given by f1 (u) = u(1 − u) = g1 (u),

f2 (u) = min(u, 1 − u) = g2 (u).

Obviously the functions fi , gi for i = 1, 2 are increasing on [0, 1/2] and decreasing on [1/2, 1]. Now we will investigate the cases that C given by (2.1) is a copula and then several selected dependence measures for C will be calculated. Clearly, the new family of copulas is given by C(u, v) = uv + λ1 uv(1 − u)(1 − v) + λ2 min(u, 1 − u) min(v, 1 − v).

(4.1)

As f1 , g1 , f2 , g2 satisfy the boundary conditions, hence our aim is in restriction of the parameters λ1 , λ2 such that C is 2-increasing. The copula density function has the following form:    c(u, v) = 1 + λ1 (1 − 2u)(1 − 2v) + λ2 1]0,1/2[ (u) − 1]1/2,1[ (u) 1]0,1/2[ (v) − 1]1/2,1[ (v)  0. (4.2) It is clear that α1 = α2 = γ1 = γ2 = −1 and β1 = β2 = δ1 = δ2 = 1. Based on Proposition 2.1 we have τ1 , τ2 ∈ [−1, 1], thus for any a ∈ [0, 1], (λ1 , λ2 ) = (aτ1 , (1 − a)τ2 ) yields a copula (see Fig. 1). (λ1 , λ2 ) is a point in convex set determined by vertices (−1, 0), (0, −1), (1, 0), (0, 1). Based on Proposition 2.3 we have to find solution of: (1) (2) (3)

(d1 , d2 ) (1, 1) (1, 0) (1, 0)

(e1 , e2 ) (0, 0) (0, 0) (0, 1)

(4)

(0, 1)

(0, 0)

(5) (6) (7)

(0, 1) (0, 0) (0, 0)

(1, 0) (1, 1) (1, 0)

(8)

(0, 0)

(0, 1)

λ1 + λ2  −1 λ1  −1 λ1 − λ2  −1 λ2  −λ1 + λ2  −λ1 − λ2  −λ1 

−1 −1 −1 −1

−λ2  −1.

(4.3)

Summarizing it should hold λ1 , λ2 , λ1 + λ2 , λ1 − λ2 ∈ [−1, 1] yielding the same area as in the previous case, see Fig. 1. With Theorem 2.2, complete solution is given below (see Fig. 1), −1  λ1 + λ2  1 −1  λ2  1.

(4.4)

For example, consider λ1 = 2, λ2 = −1. Then applying (2.1), we have C(u, v) = uv + 2uv(1 − u)(1 − v) − min(u, 1 − u) min(v, 1 − v). We depict C in Fig. 2.

(4.5)

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Fig. 2. Copula C on left and its density c on right for λ1 = 2, λ2 = −1.

Fig. 3. Possible area for (λ1 , λ2 ) based on Propositions 2.1, 2.3 (Left), and based on Theorem 2.2 (Right).

Observe that the copula C given by (4.5) cannot be obtained as a convex combination of copulas C1 , C2 constructed by means of Rodríguez-Lallena and Úbeda-Flores approach, C1 = uv + τ1 uv(1 − u)(1 − v)

(4.6)

and C2 = uv + τ2 min(u, 1 − u) min(v, 1 − v),

τ1 , τ2 ∈ [−1, 1].

(4.7)

The dependence measures introduced in Section 3 for this new family are calculated as below: 1 1 τ C = λ1 + λ2 4 2 1 3 ρ C = λ1 + λ2 3 4 4 2 γ C = λ 1 + λ2 15 3 tLC = 0 = tUC . Example 4.2. Let functions f1 (u) = u(1 − u) = g1 (u) and f2 = g2 be given as follows, ⎧ ⎨ u − 1/4 if 1/4  u  1/2 f2 (u) = 3/4 − u if 1/2  u  3/4 ⎩ 0 else.

(4.8)

(4.9)

Obviously the functions f1 , g1 are increasing on [0, 1/2] and decreasing on [1/2, 1]. Moreover, f2 , g2 are equal to 1 on the interval ]1/4, 1/2[ and −1 on ]1/2, 3/4[. Clearly C is given by C(u, v) = uv + λ1 uv(1 − u)(1 − v) + λ2 f2 (u)g2 (v).

(4.10)

As in Example 4.1, it is clear that α1 = α2 = γ1 = γ2 = −1 and β1 = β2 = δ1 = δ2 = 1. Based on Proposition 2.1 and Proposition 2.3, similar to Example 4.1, parameters are restricted by λ1 , λ2 , λ1 + λ2 , λ1 − λ2 ∈ [−1, 1]. On the other hand, via Theorem 2.2, complete solution is given below (see Fig. 3): 1 −1  λ1 + λ2  1 4 −1  λ1  1. The dependence measures introduced in Section 3 for this new family are calculated as below:

(4.11)

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Fig. 4. Possible area for (λ1 , λ2 ) based on Propositions 2.1, 2.3 (Left), and based on Theorem 2.2 (Right).

1 1 τ C = λ1 + λ 2 4 32 1 3 ρ C = λ1 + λ 2 3 64 4 1 γ C = λ 1 + λ2 15 8 tLC = 0 = tUC .

(4.12)

In Examples 4.1, 4.2 as f1 (u) = g1 (u) and f2 (u) = g2 (u), the proposed new families are symmetric modifications of the product copula. Now we consider asymmetric examples. Example 4.3. Let functions f1 , f2 and f2 , g2 be given as below,       1 1 f1 (u) = max 0, min u, − u , g1 (v) = max 0, min v − , 1 − v , 2 2     2 2 3 f2 (u) = max 0, 3u − 1 − 2u , g2 (v) = max 0, v − 2v . It is clear that α1 = α2 = γ1 = −1, β1 = β2 = δ1 = 1 and γ2 = −1/2, δ2 = 1/6. Based on Proposition 2.1 we have τ1 ∈ [−1, 1] and τ2 ∈ [−2, 2], thus for any a ∈ [0, 1], (λ1 , λ2 ) = (aτ1 , (1 − a)τ2 ) yields a copula (see Fig. 4). (λ1 , λ2 ) is a point in convex set determined by vertices (−1, 0), (0, −2), (1, 0), (0, 2). Based on Proposition 2.3 we have to find solution of: (d1 , d2 ) (e1 , e2 ) (1)

(1, 1)

(0, 0)

(2)

(1, 0)

(0, 0)

λ1 + 12 λ2  −1 λ1  −1

(3)

(1, 0)

(0, 1)

λ1 − 12 λ2  −1

(4)

(0, 1)

(0, 0)

(5)

(0, 1)

(1, 0)

−λ1 +

(6)

(0, 0)

(1, 1)

−λ1 −

(7)

(0, 0)

(1, 0)

 −1 −λ1  −1

(8)

(0, 0)

(0, 1)

− 12 λ2  −1.

1 2 λ2 1 2 λ2 1 2 λ2

 −1  −1

(4.13)

Summarizing, it should hold λ1 , λ1 + − ∈ [−1, 1] and λ2 ∈ [−2, 2], yielding the same area as in the previous case in Proposition 2.1, see Fig. 4. With Theorem 2.2, complete solution is given below (see Fig. 4), 1 λ 1 + λ2  1 2 −1  λ1  1 1 2 λ2 , λ1

−2  λ2 .

1 2 λ2

(4.14)

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Fig. 5. Possible area for (λ1 , λ2 ) based on Propositions 2.1, 2.3 (Left), and based on Theorem 2.2 (Right).

The dependence measures introduced in Section 3 for this new family are calculated as below:   λ1 λ2 τC = + 32 512 3 ρC = (λ1 + λ2 ) 64 γC = 0 tLC = 0 = tUC .

(4.15)

Example 4.4. Let functions f1 , g1 , f2 , g2 : [0, 1] → R be given by    1 1 f1 (u) = sin(3πu), g1 (v) = max 0, min v, − v , 3π 2    1 g2 (v) = max 0, min v − , 1 − v . f2 (u) = u(1 − u), 2 Obviously as f1 , g1 , f2 , g2 satisfy the boundary conditions, hence our aim is in restriction of the parameters λ1 , λ2 such that C is 2-increasing. It is clear that α1 = α2 = γ1 = γ2 = −1 and β1 = β2 = δ1 = δ2 = 1. Based on Proposition 2.1 we have τ1 , τ2 ∈ [−1, 1] thus for any a ∈ [0, 1], (λ1 , λ2 ) = (aτ1 , (1 − a)τ2 ) yields a copula (see Fig. 5). (λ1 , λ2 ) is a point in convex set determined by vertices (−1, 0), (0, −1), (1, 0), (0, 1). Based on Proposition 2.3 we have to find solution of: (1) (2) (3)

(d1 , d2 ) (1, 1) (1, 0) (1, 0)

(e1 , e2 ) (0, 0) (0, 0) (0, 1)

(4)

(0, 1)

(0, 0)

(5) (6) (7)

(0, 1) (0, 0) (0, 0)

(1, 0) (1, 1) (1, 0)

(8)

(0, 0)

(0, 1)

λ1 + λ2  −1 λ1  −1 λ1 − λ2  −1 λ2  −λ1 + λ2  −λ1 − λ2  −λ1 

−1 −1 −1 −1

(4.16)

−λ2  −1

summarizing it should hold λ2 , λ1 , λ1 − λ2 , λ1 + λ2 ∈ [−1, 1] yielding the same area as in the previous case in Proposition 2.1. With Theorem 2.2, complete solution is given below (see Fig. 5), −1  λ1  1,

−1  λ2  1.

(4.17)

Remark 4.5. If supports of the functions fi and gi are disjoint then we can apply directly the result of RodríguezLallena and Úbeda-Flores, namely for i = 1, 2, . . . , n, −1 −1  λi  . max(αi γi , βi δi ) min(αi δi , βi γi )

(4.18)

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5. Relationships to some other construction methods Our construction method (2.1) can be further discussed. For example, it is compatible with ordinal sum construction of copulas (see Nelsen [15], Mesiar and Sempi [13]). Indeed consider any ordinal sum copula   D = uk , vk , Π /k ∈ K (5.1) of product copulas, where (]uk , vk [)k∈K is a disjoint system of open subintervals of [0, 1] and D is given by  )(v−vk ) , if (u, v) ∈ ]uk , vk [2 for some k ∈ K uk + (u−uvkk−u k D(u, v) = min(u, v), else.

(5.2)

For k ∈ K, let fk , gk : [0, 1] → R be absolutely continuous functions such that the function Ck : [0, 1]2 → [0, 1] given by Ck (u, v) = uv + λk fk (u)gk (v) is a copula for any λk ∈ [pk , qk ], k ∈ K. Define for any k ∈ K the functions fk∗ , gk∗ : [0, 1] → R by  k fk ( vu−u ), if u ∈ ]uk , vk [ ∗ k −uk fk (u) = (5.3) 0, else.  k gk ( vv−v ), if v ∈ ]uk , vk [ k −uk gk∗ (v) = (5.4) 0, else. Evidently, the supports of the functions (fk∗ )k∈K are disjoint. The same claim holds for functions (gk∗ )k∈K . It is not difficult to see, compare also Remark 4.5, that the ordinal sum   C = uk , vk , Ck /k ∈ K (5.5) can be expressed in the form  C(u, v) = D(u, v) + λk fk∗ (u)gk∗ (v)

(5.6)

k∈K

and it is a copula whenever for each k ∈ K, λk ∈ [pk , qk ]. Remark 5.1. Another interesting observations concerns the relationship of our construction method and flipping/survival construction method for copulas (see Nelsen [15]). Due to the fact, that a function C : [0, 1]2 → [0, 1] is a copula if and only if also the function D : [0, 1]2 → [0, 1] given by D(u, v) = u − C(u, 1 − v)

(5.7)

is a copula (then D is said to be the second coordinate flipping of C), we see that if a function C given by (2.1) is a copula, then also the corresponding function D given by D(u, v) = uv −

n 

λi fi (u)gi (1 − v)

(5.8)

i=1

is a copula (and vice-versa). A similar claim holds for the first coordinate flipping of C which we denote by E, and the survival copula F related to C, which are given by E(u, v) = uv −

n 

λi fi (1 − u)gi (v)

(5.9)

i=1

and F (u, v) = uv +

n 

λi fi (1 − u)gi (1 − v).

(5.10)

i=1

Evidently, for the functions systems ((fi (u), gi (v))/i = 1, . . . , n), ((−fi (1 − u), gi (v))/i = 1, . . . , n), ((fi (u), −gi (1 − v))/i = 1, . . . , n) and ((−fi (1 − u), −gi (1 − v))/i = 1, . . . , n) the domains of parameters (λ1 , . . . , λn ) turning the function given in (2.1) to be a copula coincide for all these four functions systems.

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Finally, recall a generalization of flipping/survival constructions of copulas based on a couple of parameters (a, b) from [0, 1]2 introduced in Nelsen [15], relating to any copula C : [0, 1]2 → [0, 1] a (possibly new) copula C(a,b) : [0, 1]2 → [0, 1] given by means of volumes of copula C as follows: 



 C(a,b) (u, v) = VC a(1 − u), u + a(1 − u) × b(1 − v), v + b(1 − v) . For more details we refer the reader to the related study by De Baets et al. [2]. It is not difficult to check that, for a copula C given by (2.1), it holds then C(a,b) (u, v) = uv +

n 

  λi fi,a (u)gi,b (v)

(5.11)

i=1

where     fi,a (u) = fi u + a(1 − u) − fi a(1 − u)

(5.12)

    gi,b (v) = gi v + b(1 − v) − gi b(1 − v) .

(5.13)

and

Hence, if a copula C given by (2.1) is linked to a functions system ((fi , gi )/i = 1, . . . , n) then the copula C(a,b) is linked to the functions system 

 (fi,a , gi,b )/i = 1, . . . , n .

(5.14)

Obviously, the domain of parameters (λ1 , . . . , λn ) related to the first functions system turning (2.1) into a copula is a subset (which is proper, in general) of the domain related to the second functions system. As an example, consider the functions system described in Example 4.1, see also Fig. 1 for the discussed parameters domains. Taking into account a = b = 0.5, all considered functions after the corresponding transformation turn to the null functions, and thus the new parameters domain has no restriction, for each (λ1 , λ2 ) from R2 the formula (2.1) yields the product copula C(u, v) = uv. Example 5.2. Let f (u) = u(1 − u). Then we get fa (u) = u(1 − u)(1 − 2a). And if we let f (u) = min(u, 1 − u), then we get   fa (u) = sign(1 − 2a) min u, (1 − u)(1 − 2a) . Remark 5.3. In general, fa (u) = f (u + a(1 − u)) − f (a(1 − u)) and therefore fa (u) = (1 − a)f  (u + a(1 − u)) + af  (a(1 − u)). It is clear that fa (u) ∈ [α, β], where α, β are the same as in (1.2). 6. Conclusion In this study, we have suggested new parametric symmetric/asymmetric families of copulas via modification of the product copula. The extended new families are more flexible than the original family and are able to model the more miscellaneous structures. Several selected dependence measures such as Kendall’s tau, Spearman’s rho, Gini’s gamma of these new families were investigated and at last several examples were shown. Our approach has interesting relationship with some other construction methods for copulas. We have discussed these links in the case of standard ordinal sums and volume-based construction from Nelsen [15]. Similar relationships are expected for each construction method preserving the piecewise bilinearity of copulas, such as W - or P -based ordinal sums, see Mesiar and Szolgay [14], De Baets and De Meyer [1] and also Mesiar et al. [11]. Moreover, our approach can be adapted to the multivariate copulas, too, thus generalizing the results of Dolati and Úbeda-Flores from [3].

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Acknowledgements The work on this paper was realized during the Erasmus stage of the second author at STU/Bratislava, and it was supported by the grants APVV-0073-10 and by the European Regional Development Fund in the IT4 Innovations Centre of Excellence Project (C.Z.1.05/1.1.00/02.0070). The authors are grateful to the anonymous referees and editors whose deep comments helped to improve the original version of this contribution. References [1] B. De Baets, H. De Meyer, Orthogonal grid constructions of copulas, IEEE Trans. Fuzzy Syst. 15 (2007) 1053–1062. [2] B. De Baets, H. De Meyer, J. Kalická, R. Mesiar, Flipping and cyclic shifting of binary aggregation functions, Fuzzy Sets Syst. 160 (2009) 752–765. [3] A. Dolati, M. Úbeda-Flores, Some new parametric families of multivariate copulas, Int. Math. Forum 1 (2006) 17–25. [4] F. Durante, P. Jaworski, Invariant dependence structure under univariate truncation, Statistics 46 (2012) 263–267. [5] F. Durante, A. Kolesárová, R. Mesiar, C. Sempi, Copulas with given diagonal sections: novel constructions and applications, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 15 (4) (2007) 397–410. [6] F. Durante, A. Kolesárová, R. Mesiar, C. Sempi, Copulas with given values on a horizontal and a vertical section, Kybernetika 43 (2) (2007) 209–220. [7] F. Durante, A. Kolesárová, R. Mesiar, C. Sempi, Semilinear copulas, Fuzzy Sets Syst. 159 (1) (2008) 63–76. [8] N.I. Fisher, Copulas, in: S. Kotz, C.B. Read, D.L. Banks (Eds.), Encyclopedia of Statistical Sciences, vol. 1, Wiley, New York, 1997, pp. 159–163. [9] T. Jwaid, B. De Baets, J. Kalická, R. Mesiar, Conic aggregation functions, Fuzzy Sets Syst. 167 (2011) 3–20. [10] 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. [11] R. Mesiar, V. Jágr, M. Juráˇnová, M. Komorníková, Univariate conditioning of copulas, Kybernetika 44 (6) (2008) 807–816. [12] R. Mesiar, J. Komorník, M. Komorníková, On some construction methods for bivariate copulas, in: Aggregation Functions in Theory and in Practise, Springer-Verlag, Berlin, Heidelberg, 2013, pp. 39–45. [13] R. Mesiar, C. Sempi, Ordinal sums and idempotents of copulas, Aequ. Math. 79 (2010) 39–52. [14] R. Mesiar, J. Szolgay, W-ordinal sums of copulas and quasi-copulas, in: Proc. MAGIA Conference, Koˇcovce, 2004, pp. 78–83. [15] R.B. Nelsen, An Introduction to Copulas, second edition, Springer, New York, 2006. [16] J.A. Rodríguez-Lallena, M. Úbeda-Flores, A new class of bivariate copulas, Stat. Probab. Lett. 66 (2004) 315–325. [17] O.V. Sarmanov, Generalized normal correlation and two-dimensional Fréchet classes, Dokl. Akad. Nauk SSSR 168 (1966) 32–35 (in Russian). [18] A. Sklar, Fonctions de répartition á n dimensions et leurs marges, Publ. Inst. Stat. Univ. Paris 8 (1959) 229–231.