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A note on Archimax copulas and their representation by means of conic copulas
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Short Communication
A note on Archimax copulas and their representation by means of conic copulas Michal Dibala, Lucia Vavríková ∗ Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia Received 22 February 2016; received in revised form 3 June 2016; accepted 7 June 2016
Abstract The aim of this note is to prove the equivalence between the class of conic copulas and the class of Archimax copulas based on the lower Fréchet–Hoeffding bound W . We demonstrate the transition from arbitrary conic copula to appropriate Pickands dependence function, and vice-versa. Our results show that Archimax copulas are isomorphic transforms of either EV -copulas or conic copulas. © 2016 Elsevier B.V. All rights reserved. Keywords: Aggregation function; Archimax copula; Conic copula; Dependence function
1. Introduction In many areas of practice we are encountered with the objective to model relationships between random variables. Copula theory is one of possible approaches to solve this problem. At the present time, there exist a huge quantity of copula classes, and so it is problematic also for experts to choose the right one. Sometimes it is possible to express the same copula in different forms coming from different copula classes. This is, e.g., the case of Clayton copulas having different representations as Archimedean copulas [11] and as univariate conditioning stable copulas [3]. During our studies of piecewise linear copulas we observed a link between conic copulas characterized by a convex decreasing function h: [0, 1] → [0, 1], see [2], and Archimax copulas based on the Fréchet–Hoeffding lower bound W , characterized by a dependence function D: [0, 1] → [0, 1]. The aim of this contribution is a complete description of this correspondence, including its generalization for quasi-copulas. The paper is organized as follows. In Section 2 we comprehend necessary knowledge about conic and Archimax copulas. Then, in the third section, we prove the equivalence between the class of conic copulas and Archimax copulas based on W and we introduce examples of the transition from Archimax copula class to conic copula class, and vice-versa. In Section 4, some additional results are included. Namely, the class of all Archimax copulas is shown to * Corresponding author.
E-mail addresses: [email protected] (M. Dibala), [email protected] (L. Vavríková). http://dx.doi.org/10.1016/j.fss.2016.06.003 0165-0114/© 2016 Elsevier B.V. All rights reserved.
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be the class of all copulas which are isomorphic either to extreme value copulas [11] or to conic copulas. Next, we extend the coincidence of classes of W -based Archimax copulas and of conic copulas to the class of quasi-copulas. Finally, some concluding remarks are added. 2. Basic definitions and preliminaries Let I = [0, 1]. Following [1,4] we introduce basic definitions and preliminaries of copulas. Definition 2.1. Let f : I → [0, ∞] be a convex continuous strictly decreasing function fulfilling condition f (1) = 0. Then the function f (−1) : [0, ∞] → I , f (−1) (t) = min (t, f (0)) is a pseudo-inverse of f . Theorem 2.1. Let f fulfill conditions in Definition 2.1. Then the function Cf : I 2 → I , Cf (x, y) = f (−1) (f (x) + f (y)) is a (two-dimensional) copula. Recall that copulas Cf are called Archimedean and the related function f is called an additive generator (of Cf ). Moreover, Cf = Cg if and only if f = c · g for some positive constant c. For more details we recommend [11]. Two-dimensional Archimax copulas were introduced by Capéraà, Fougères and Genest [2] as a common extension of both extreme-value copulas and Archimedean copulas. Theorem 2.2. Let f be an additive generator of some Archimedean copula and let D : I → [ 12 , 1] be a Pickands dependence function, i.e., a convex function bounded from below by max(z, 1 − z). Then the function Cf,D : I 2 → I given by f (x) Cf,D (x, y) = f (−1) (f (x) + f (y)) · D (1) f (x) + f (y) is a copula. Copula Cf,D from the above theorem is called an Archimax copula based on the additive generator f and Pickands dependence function D. If f (x) = − ln x we get the class of extreme-value copulas [2]. Obviously, if D(z) = 1 for all z ∈ I then Cf,D = Cf is an Archimedean copula. On the other side, if D(z) = max(z, 1 − z) then Cf,D = M, independently of the additive generator f . Here M: I 2 → I is the upper Fréchet–Hoeffding bound, M(x, y) = min(x, y). In general Cf ≤ Cf,D ≤ M. Another recently introduced class of copulas is formed by conic copulas [8]. Definition 2.2. Let Z = (x, y) ∈ I 2 |0 ∈ {x, y} . A set Z ⊂ I 2 (Z = I 2 ), Z ⊆ Z is called a zero set, if it is closed and if x ∈ Z and u ∈ I 2 satisfy u ≤ x then u ∈ Z, where u = (u1 , u2 ), x = (x1 , x2 ) and u ≤ x means u1 ≤ x1 and u2 ≤ x2 . Zero set ZA of a continuous aggregation function A : I 2 → I with annihilator a = 0, see [6], is the inverse image of the value 0, i.e., ZA := A−1 ({0}) = x ∈ I 2 |A(x) = 0 , (2) and clearly Z ⊂ ZA . Since A(1, 1) = 1, ZA is a proper subset of I 2 . A point x = (x1 , x2 ) ∈ ZA is called weakly undominated point if there exists no y = (y1 , y2 ) ∈ ZA such that y1 > x1 and y2 > x2 . We will refer to the set of all weakly undominated points of zero-set ZA of a continuous aggregation function A as the upper boundary curve of ZA [7]. Now a general definition of conic functions follows. We denote the (linear) segment with endpoints x, y ∈ I 2 as
x, y = {λx + (1 − λ)y | λ ∈ I } . Definition 2.3. [8] Let Z ⊂ I 2 be a zero set. We define the function AZ : I 2 → I as follows: • AZ (1) = 1,
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• AZ (x) = 0 for any x ∈ Z, • for any weakly undominated point x ∈ Z, the function AZ is linear on the segment x, 1 . The function AZ is called a conic function generated by a zero-set Z. Theorem 2.3. [7] Let Z be a zero set such that Z = Z ∪ {x ∈ I 2 | x1 ≤ d, x2 ≤ h(x1 )} induced by a continuous function h : [0, d] → I , d ∈ I . The conic function AZ is a copula if and only if the function h satisfies the following conditions: • h(d) = 0, • h is convex. Observe that AZ = M and AZ = W , where the zero set Z is given by Z = {x ∈ I 2 | x1 + x2 ≤ 1}. Copulas introduced in Theorem 2.3 are called conic copulas and we adopt notation C[h] for them. For some further details on conic copulas see also [5,9]. Recall that conic copulas are piece-wise linear on any segment which is an intersection of the domain I 2 with a straight line containing the vertex 1. However, a similar property can be observed in the case of Archimax copulas linked to the Fréchet–Hoeffding lower bound W , W (x, y) = max(x + y − 1, 0). Recall that W = Cg = C[g] , where g: I → I is given by g(x) = 1 − x. Therefore we are interested in Archimax copulas based on W copula. From equation (1) we get 1−x Cg,D (x, y) = max 0, 1 − (2 − x − y) · D . (3) 2−x −y Recall again that for the greatest dependence function D ∗ given by D ∗ (z) = 1, z ∈ I , the formula (3) yields Cg,D ∗ (x, y) = max (0, 1 − (2 − x − y) · 1) = max(0, x + y − 1) = W (x, y) = AZ (x, y). On the contrary, when we consider the smallest dependence function D∗ given by D∗ (z) = max(z, 1 − z), then we get Cg,D∗ (x, y) = min(x, y) = M(x, y) = AZ (x, y). 3. Representation of Archimax copulas by means of conic copulas A natural question arises: what is the relationship of the classes of W -related Archimax copulas and of conic copulas? In this section we prove that there is an equivalence between the conic copula class and the Archimax copula class related to W . Theorem 3.1. The class of conic copulas is equivalent to the class of Archimax copulas based on W . Proof. We split the proof into two implications. First consider Cg,D : I 2 → I be an Archimax copula related to W . We have to show that there is a convex function h : I → I such that Cg,D is a conic copula, Cg,D = C[h] . Based on the formula (3) it is evident that the zero set ZCg,D related to the Archimax copula Cg,D is given by 1−x 2 ZCg,D = (x, y) ∈ I | 1 ≤ (2 − x − y) · D . 2−x −y Denote aD = max{z ∈ I | D(z) = 1 − z} and bD = min{z ∈ I | D(z) = z}. Then 0 ≤ aD ≤ 12 ≤ bD ≤ 1 and Cg,D (x, y) = y if and only if y≤
(1 − aD )x + 2aD − 1 0 (with convention = 1). aD 0
Similarly Cg,D (x, y) = x if and only if y ≥
(1 − bD )x + 2bD − 1 . bD
Thus, for z ∈ [0, aD ] ∪ [bD , 1], the points (x, y) ∈ I 2 satisfying
1−x = z and (x, y) ∈ ZCg,D belong to Z . 2−x −y
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1−x . Then (x, y) is a weakly undominated point of ZCg,D if and only if 2−x −y 1−x 1 = (2 − x − y) · D 2−x −y or, equivalently z = (1 − x) · D(z). Then z 1−z x =1− and y = 1 − . (4) D(z) D(z) Due to the fact that for any z1 , z2 ∈ ]aD , bD [, |D(z1 ) − D(z2 )| < |z1 − z2 |, it follows that (4) determines a function 1 − 2aD 2bD − 1 h: [0, d] → [0, 1], h(x) = y, where d = , h(d) = 0, and h(0) = . 1 − aD bD Now, it is a matter of an easy processing to see that Cg,D is a conic aggregation function related to the zero set ZCg,D . Due to the fact that Cg,D is a copula, it follows that Cg,D = C[h] , i.e., Cg,D is a conic copula and h is a convex function. The second implication considers C[h] : I 2 → I , a conic copula. We have to show that then there is a dependence function D : I → I such that C[h] = Cg,D , i.e., C[h] is an Archimax copula related to W . 1 − C[h] (x, y) As will be shown in Example 3.2, for z ∈]0, 1[ it is enough to define D(z) = for an arbitrary point 2−x −y 1−x (x, y) ∈]0, 1[2 such that C[h] (x, y) > 0 and z = . Then D is well defined due to the linearity of conic copula 2−x −y C[h] on segment determined by the above constraints. Setting D(0) = D(1) = 1, D is a well defined continuous function, D : I → I . Applying formula (1) on Cg,D it holds Cg,D = C[h] . From the fact that Cg,D is a copula it follows that D is a Pickands dependence function. 2 Consider z ∈ ]aD , bD [, z =
In the following part we show an example in which for a chosen dependence function D of an Archimax copula we find the corresponding conic copula. Example 3.1. Let D(z) = z2 − z + 1, z ∈ I . Then D is a convex function and max(1 − z, z) ≤ D(z) ≤ 1, i.e., D is Pickands dependence function (Fig. 1). From (3) we get
2 1−x 1−x Cg,D (x, y) = max 0, 1 − (2 − x − y) · − +1 2−x −y 2−x −y x 2 + y 2 + xy − 2y − 2x + 1 = max 0, − . (5) 2−x −y Then from expression x 2 + y 2 + xy − 2y − 2x + 1 =0 2−x −y we obtain the formula for the margin of the considered zero set in the next form: y 2 + (x − 2)y + x 2 − 2x + 1 = 0. The only proper solution is: 3 x h(x) = 1 − − x − x 2 . 2 4 Its second derivative is given by: 1 − d 2 h(x) 3 = > 0, ∀x ∈]0, 1[. dx 2 3 2 4 x x− x x− 3 4
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Fig. 1. Pickands dependence function D and Archimax copula Cg,D .
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Fig. 2. Pickands dependence function D.
Because the second derivative is positive on ]0, 1[, the function h is convex on I and it really describes the zero set ZCg,D . To show that Cg,D is a conic copula, we apply a parametrization y = ax + b, a > 0. From the fact that copula Cg,D has to be linear on segments containing the point (1, 1), we have ax + b = 1, x = 1 and from this we have b = 1 − a. From (5) we see the piecewise linearity of copula Cg,D on the corresponding segment, namely (a 2 + a + 1)x 2 − (2a 2 + a + 1)x + a 2 (a 2 + a + 1)x − a 2 Cg,D (x, y) = max 0, = max 0, . (x − 1)(a + 1) a+1 Consequently, copula Cg,D is linear on the maximal segment of the straight line y = cx + 1 − c (where c = 1 a + a+1 ), which is a subset of I 2 . We have demonstrated the linearity of the copula Cg,D on linear segments connecting the points of the zero set margin with the point (1, 1, 0) and convexity of the function h describing the zero set margin, i.e., Cg,D = C[h] . In the following example we show the transition from the class of conic copulas to the class of Archimax copulas for chosen conic copula.
Example 3.2. Let h : 0, 12 → I be given by h(x) = (1 − 2x)2 . Obviously, h satisfies all constraints of Theorem 2.3. Then the conic copula C[h] (Figs. 2, 3) is defined as follows [8]: ⎧ 0 0 ≤ x ≤ 12 , y ≤ (1 − 2x)2 , ⎪ ⎪ ⎪ ⎪ ⎨ C[h] (x, y) = 4x(1 − x) − 1 + y (1 − 2x)2 < y and y ≥ 2x − 1, ⎪ 4(1 − x) − 1 + y ⎪ ⎪ ⎪ ⎩ min(x, y) else. Our aim is to represent the conic copula C[h] in the Archimax form Cg,D , see formula (3). 1−x For (x, y) ∈]0, 1[2 , z = , then for a chosen constant value z ∈ ]0, 1[ it holds: 2−x −y y=
1−z 2z − 1 x+ , z z
z = 0.
From (3) we obtain the formula D(z) =
(6) 1 − C[h] (x, y) describing a dependence function D. 2−x −y
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Fig. 3. Conic copula C[h] .
From (3) and (6) we get:
D(z) =
⎧ ⎪ ⎪ 1−z ⎪ ⎨ ⎪ ⎪ 4z2 ⎪ ⎩ 5z − 1
1 , z ∈ 0, 3 1 z ∈ ,1 . 3
The function D(z) is continuous and we show its convexity. 1 1 The function D1 = D| 0, is given by D1 (z) = 1 − z, z ∈ 0, . It is linear and therefore convex, and 3 3 − 1 dD1 3 = −1. dz 1 4z2 8 d 2 D2 (z) The function D2 = D| , 1 , D2 (z) = = > 0, and thus D2 is is twice differentiable, 2 3 5z − 1 dz (5z − 1)3 + 1 dD2 3 = −1, therefore the function D is convex on I and fulfills all conditions of Pickands convex. Moreover dz dependence function in definition of Archimax copulas. 4. Some other remarks Our previous results have an interesting consequence that Archimax copulas are isomorphic transforms of EV -copulas and of conic copulas. Observe that due to the isomorphism φ : I → I of a fixed (but arbitrarily chosen) non-strict Archimedean copula C and W (i.e., φ is a concave automorphism and f = 1 − φ = g ◦ φ is an additive generator of C), for the related Archimax copulas it holds Cf,D (x, y) = f =φ
−1
(−1)
(f (x) + f (y)) · D
f (x) f (x) + f (y)
1 − φ(x) max (0, (2 − φ(x) − φ(y)) · D 2 − φ(x) − φ(y)
,
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Fig. 4. Dependence functions D1 and D2 .
i.e., φ is an isomorphism between Archimax copulas Cf,D and Cg,D . Therefore, any Archimax copula Cf,D based on an additive generator f of a non-strict Archimedean copula C can be seen as a φ-transform of a conic copula Cf,D (x, y) = φ −1 C[h] (φ(x), φ(y)) . A similar relation is valid for the case of strict copulas. In fact, they are isomorphic to the product copula, and hence each Archimax copula Cf,D , such that Cf is a strict copula, is isomorphic to an extreme value copula linked to the Pickands dependence function D. Summarizing, we have the next observation: a copula C is an Archimax copula if and only if it is either isomorphic to some EV -copula [10], or to some conic copula. Another interesting result concerns the relation between conic quasi-copulas further studied in [8] and Archimax quasi-copulas, which can be defined similarly to Archimax copulas as follows. Definition 4.1. Let f : I → R be a function and let z0 ∈ I be fixed. We say that f is convex in point z0 if for any y, λ ∈ I it holds f (λz0 + (1 − λ) y) ≤ λf (z0 ) + (1 − λ) f (y) . Definition 4.2. Let f be an additive generator of some Archimedean copula and let D : I → I be a function convex in points z1 = 0 and z2 = 1, bounded from below by max (z, 1 − z). Then the function Qf,D : I 2 → I given by f (x) Qf,D (x, y) = f (−1) (f (x) + f (y)) · D f (x) + f (y) is called an Archimax quasi-copula. Example 4.1. Let D1 (z) = max −z + 1, 3z + 23 and D2 (z) = max − 3z + 1, z be two Pickands dependence functions, see Fig. 4. Then the function D (z) = min (D1 (z) , D2 (z)) is obviously not convex, but it is convex in points 0 and 1. 1−x It is easy to see that the function Qg,D given by Qg,D (x, y) = 1 − (2 − x − y) · D 2−x−y is a quasi-copula but not a copula. Indeed Qg,D (0, 0) = 0, Q Qg,D (x,1) = x and (1, y)= y, but themargin of the Qg,D g,D (1,1) =1,
zero set is composed of line segments (0, 1) , 0, 23 , 0, 23 , 25 , 25 , 25 , 25 , 23 , 0 and 23 , 0 , (1, 0) . The margin of the zero set is not convex (see Fig. 5), therefore the function Qg,D (see Fig. 6) is not a copula, but it fulfills all conditions of quasi-copula, i.e., Qg,D is a proper quasi-copula.
Because of [11] we know that supremum of every non-empty set of copulas is a quasi-copula. Moreover, any function D characterized in Definition 4.2 can be obtained as an infimum of Pickands dependence functions, D =
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Fig. 5. Dependence function D and margin of zero set ZQg,D .
Fig. 6. Quasi-copula Qg,D .
inf{Dt | t ∈ T } and thus any W -based Archimax quasi-copula Qg,D is a quasi-copula, Qg,D = sup{Cg,Dt | t ∈ T } = sup{C[ht ] | t ∈ T } = Q[ht ] , where ht = inf{ht | t ∈ T }. For the definition of conic quasi-copulas Q[h] see [7,8]. 5. Conclusion We have shown the equality of classes of conic copulas and of Archimax copulas related to the weakest copula W , including a similar result for quasi-copulas. This class ranges from W to M and thus we expect its applications in fitting copulas to real world problems, where there is no a priori known constraint on dependence parameters, such as Kendall’s τ or Spearman’s ρ. Moreover, we have stressed the characterization of the class of all Archimax copulas as isomorphic transforms of either EV -copulas or conic copulas. To stress the impact of our result, note that the computation of Spearman’s ρ for a W -based Archimax copula Cg,D given by (3) can be rather complicated. However, if we express Cg,D as a conic copula, Cg,D = C[h] , then it is not difficult to check that
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ρCg,D = ρC[h]
⎞ ⎛ d d 1 ⎝ ⎠ = 12 · · 1 − h(x) dx − 3 = 1 − 4 h(x) dx. 3 0
0
Acknowledgement This work was supported by grants VEGA 1/0420/15 and APVV 14-0013. The authors express their gratitude to anonymous referees for all comments which have helped them in improving the original version of this paper. References [1] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Stud. Fuzziness Soft Comput., vol. 221, Springer, Berlin, 2007. [2] P. Capéraà, A.-L. Fougères, C. Genest, Bivariate distributions with given extreme value attractor, J. Multivar. Anal. 72 (2000) 30–49. [3] F. Durante, P. Jaworski, Invariant dependence structure under univariate truncation, Statistics 46 (2) (2012) 263–277. [4] F. Durante, C. Sempi, Principles of Copula Theory, Chapman & Hall, London, 2015. [5] J. Fernandéz-Sanchéz, M. Úbeda-Flores, The distribution of the probability mass of conic copulas, Fuzzy Sets Syst. 284 (2016) 138–145. [6] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Encycl. Math. Appl., vol. 127, Cambridge University Press, 2009. [7] T. Jwaid, Semilinear and semiquadratic conjunctive aggregation function, PhD Thesis, Faculty of Sciences, Ghent University, Ghent, Belgium, 2014. [8] T. Jwaid, B. De Baets, J. Kalická, R. Mesiar, Conic aggregation functions, Fuzzy Sets Syst. 167 (2011) 3–20. [9] T. Jwaid, B. De Baets, H. De Meyer, The role of generalized convexity in conic copula constructions, J. Math. Anal. Appl. 425 (2015) 864–885. [10] R. Mesiar, V. Jágr, d-dimensional dependence functions and archimax copulas, Fuzzy Sets Syst. 228 (2013) 78–87. [11] R.B. Nelsen, An Introduction to Copulas, Springer, NY, ISBN 978-0-387-28659-4, 2006.
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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Short Communication
A note on Archimax copulas and their representation by means of conic copulas Michal Dibala, Lucia Vavríková ∗ Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia Received 22 February 2016; received in revised form 3 June 2016; accepted 7 June 2016
Abstract The aim of this note is to prove the equivalence between the class of conic copulas and the class of Archimax copulas based on the lower Fréchet–Hoeffding bound W . We demonstrate the transition from arbitrary conic copula to appropriate Pickands dependence function, and vice-versa. Our results show that Archimax copulas are isomorphic transforms of either EV -copulas or conic copulas. © 2016 Elsevier B.V. All rights reserved. Keywords: Aggregation function; Archimax copula; Conic copula; Dependence function
1. Introduction In many areas of practice we are encountered with the objective to model relationships between random variables. Copula theory is one of possible approaches to solve this problem. At the present time, there exist a huge quantity of copula classes, and so it is problematic also for experts to choose the right one. Sometimes it is possible to express the same copula in different forms coming from different copula classes. This is, e.g., the case of Clayton copulas having different representations as Archimedean copulas [11] and as univariate conditioning stable copulas [3]. During our studies of piecewise linear copulas we observed a link between conic copulas characterized by a convex decreasing function h: [0, 1] → [0, 1], see [2], and Archimax copulas based on the Fréchet–Hoeffding lower bound W , characterized by a dependence function D: [0, 1] → [0, 1]. The aim of this contribution is a complete description of this correspondence, including its generalization for quasi-copulas. The paper is organized as follows. In Section 2 we comprehend necessary knowledge about conic and Archimax copulas. Then, in the third section, we prove the equivalence between the class of conic copulas and Archimax copulas based on W and we introduce examples of the transition from Archimax copula class to conic copula class, and vice-versa. In Section 4, some additional results are included. Namely, the class of all Archimax copulas is shown to * Corresponding author.
E-mail addresses: [email protected] (M. Dibala), [email protected] (L. Vavríková). http://dx.doi.org/10.1016/j.fss.2016.06.003 0165-0114/© 2016 Elsevier B.V. All rights reserved.
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be the class of all copulas which are isomorphic either to extreme value copulas [11] or to conic copulas. Next, we extend the coincidence of classes of W -based Archimax copulas and of conic copulas to the class of quasi-copulas. Finally, some concluding remarks are added. 2. Basic definitions and preliminaries Let I = [0, 1]. Following [1,4] we introduce basic definitions and preliminaries of copulas. Definition 2.1. Let f : I → [0, ∞] be a convex continuous strictly decreasing function fulfilling condition f (1) = 0. Then the function f (−1) : [0, ∞] → I , f (−1) (t) = min (t, f (0)) is a pseudo-inverse of f . Theorem 2.1. Let f fulfill conditions in Definition 2.1. Then the function Cf : I 2 → I , Cf (x, y) = f (−1) (f (x) + f (y)) is a (two-dimensional) copula. Recall that copulas Cf are called Archimedean and the related function f is called an additive generator (of Cf ). Moreover, Cf = Cg if and only if f = c · g for some positive constant c. For more details we recommend [11]. Two-dimensional Archimax copulas were introduced by Capéraà, Fougères and Genest [2] as a common extension of both extreme-value copulas and Archimedean copulas. Theorem 2.2. Let f be an additive generator of some Archimedean copula and let D : I → [ 12 , 1] be a Pickands dependence function, i.e., a convex function bounded from below by max(z, 1 − z). Then the function Cf,D : I 2 → I given by f (x) Cf,D (x, y) = f (−1) (f (x) + f (y)) · D (1) f (x) + f (y) is a copula. Copula Cf,D from the above theorem is called an Archimax copula based on the additive generator f and Pickands dependence function D. If f (x) = − ln x we get the class of extreme-value copulas [2]. Obviously, if D(z) = 1 for all z ∈ I then Cf,D = Cf is an Archimedean copula. On the other side, if D(z) = max(z, 1 − z) then Cf,D = M, independently of the additive generator f . Here M: I 2 → I is the upper Fréchet–Hoeffding bound, M(x, y) = min(x, y). In general Cf ≤ Cf,D ≤ M. Another recently introduced class of copulas is formed by conic copulas [8]. Definition 2.2. Let Z = (x, y) ∈ I 2 |0 ∈ {x, y} . A set Z ⊂ I 2 (Z = I 2 ), Z ⊆ Z is called a zero set, if it is closed and if x ∈ Z and u ∈ I 2 satisfy u ≤ x then u ∈ Z, where u = (u1 , u2 ), x = (x1 , x2 ) and u ≤ x means u1 ≤ x1 and u2 ≤ x2 . Zero set ZA of a continuous aggregation function A : I 2 → I with annihilator a = 0, see [6], is the inverse image of the value 0, i.e., ZA := A−1 ({0}) = x ∈ I 2 |A(x) = 0 , (2) and clearly Z ⊂ ZA . Since A(1, 1) = 1, ZA is a proper subset of I 2 . A point x = (x1 , x2 ) ∈ ZA is called weakly undominated point if there exists no y = (y1 , y2 ) ∈ ZA such that y1 > x1 and y2 > x2 . We will refer to the set of all weakly undominated points of zero-set ZA of a continuous aggregation function A as the upper boundary curve of ZA [7]. Now a general definition of conic functions follows. We denote the (linear) segment with endpoints x, y ∈ I 2 as
x, y = {λx + (1 − λ)y | λ ∈ I } . Definition 2.3. [8] Let Z ⊂ I 2 be a zero set. We define the function AZ : I 2 → I as follows: • AZ (1) = 1,
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• AZ (x) = 0 for any x ∈ Z, • for any weakly undominated point x ∈ Z, the function AZ is linear on the segment x, 1 . The function AZ is called a conic function generated by a zero-set Z. Theorem 2.3. [7] Let Z be a zero set such that Z = Z ∪ {x ∈ I 2 | x1 ≤ d, x2 ≤ h(x1 )} induced by a continuous function h : [0, d] → I , d ∈ I . The conic function AZ is a copula if and only if the function h satisfies the following conditions: • h(d) = 0, • h is convex. Observe that AZ = M and AZ = W , where the zero set Z is given by Z = {x ∈ I 2 | x1 + x2 ≤ 1}. Copulas introduced in Theorem 2.3 are called conic copulas and we adopt notation C[h] for them. For some further details on conic copulas see also [5,9]. Recall that conic copulas are piece-wise linear on any segment which is an intersection of the domain I 2 with a straight line containing the vertex 1. However, a similar property can be observed in the case of Archimax copulas linked to the Fréchet–Hoeffding lower bound W , W (x, y) = max(x + y − 1, 0). Recall that W = Cg = C[g] , where g: I → I is given by g(x) = 1 − x. Therefore we are interested in Archimax copulas based on W copula. From equation (1) we get 1−x Cg,D (x, y) = max 0, 1 − (2 − x − y) · D . (3) 2−x −y Recall again that for the greatest dependence function D ∗ given by D ∗ (z) = 1, z ∈ I , the formula (3) yields Cg,D ∗ (x, y) = max (0, 1 − (2 − x − y) · 1) = max(0, x + y − 1) = W (x, y) = AZ (x, y). On the contrary, when we consider the smallest dependence function D∗ given by D∗ (z) = max(z, 1 − z), then we get Cg,D∗ (x, y) = min(x, y) = M(x, y) = AZ (x, y). 3. Representation of Archimax copulas by means of conic copulas A natural question arises: what is the relationship of the classes of W -related Archimax copulas and of conic copulas? In this section we prove that there is an equivalence between the conic copula class and the Archimax copula class related to W . Theorem 3.1. The class of conic copulas is equivalent to the class of Archimax copulas based on W . Proof. We split the proof into two implications. First consider Cg,D : I 2 → I be an Archimax copula related to W . We have to show that there is a convex function h : I → I such that Cg,D is a conic copula, Cg,D = C[h] . Based on the formula (3) it is evident that the zero set ZCg,D related to the Archimax copula Cg,D is given by 1−x 2 ZCg,D = (x, y) ∈ I | 1 ≤ (2 − x − y) · D . 2−x −y Denote aD = max{z ∈ I | D(z) = 1 − z} and bD = min{z ∈ I | D(z) = z}. Then 0 ≤ aD ≤ 12 ≤ bD ≤ 1 and Cg,D (x, y) = y if and only if y≤
(1 − aD )x + 2aD − 1 0 (with convention = 1). aD 0
Similarly Cg,D (x, y) = x if and only if y ≥
(1 − bD )x + 2bD − 1 . bD
Thus, for z ∈ [0, aD ] ∪ [bD , 1], the points (x, y) ∈ I 2 satisfying
1−x = z and (x, y) ∈ ZCg,D belong to Z . 2−x −y
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1−x . Then (x, y) is a weakly undominated point of ZCg,D if and only if 2−x −y 1−x 1 = (2 − x − y) · D 2−x −y or, equivalently z = (1 − x) · D(z). Then z 1−z x =1− and y = 1 − . (4) D(z) D(z) Due to the fact that for any z1 , z2 ∈ ]aD , bD [, |D(z1 ) − D(z2 )| < |z1 − z2 |, it follows that (4) determines a function 1 − 2aD 2bD − 1 h: [0, d] → [0, 1], h(x) = y, where d = , h(d) = 0, and h(0) = . 1 − aD bD Now, it is a matter of an easy processing to see that Cg,D is a conic aggregation function related to the zero set ZCg,D . Due to the fact that Cg,D is a copula, it follows that Cg,D = C[h] , i.e., Cg,D is a conic copula and h is a convex function. The second implication considers C[h] : I 2 → I , a conic copula. We have to show that then there is a dependence function D : I → I such that C[h] = Cg,D , i.e., C[h] is an Archimax copula related to W . 1 − C[h] (x, y) As will be shown in Example 3.2, for z ∈]0, 1[ it is enough to define D(z) = for an arbitrary point 2−x −y 1−x (x, y) ∈]0, 1[2 such that C[h] (x, y) > 0 and z = . Then D is well defined due to the linearity of conic copula 2−x −y C[h] on segment determined by the above constraints. Setting D(0) = D(1) = 1, D is a well defined continuous function, D : I → I . Applying formula (1) on Cg,D it holds Cg,D = C[h] . From the fact that Cg,D is a copula it follows that D is a Pickands dependence function. 2 Consider z ∈ ]aD , bD [, z =
In the following part we show an example in which for a chosen dependence function D of an Archimax copula we find the corresponding conic copula. Example 3.1. Let D(z) = z2 − z + 1, z ∈ I . Then D is a convex function and max(1 − z, z) ≤ D(z) ≤ 1, i.e., D is Pickands dependence function (Fig. 1). From (3) we get
2 1−x 1−x Cg,D (x, y) = max 0, 1 − (2 − x − y) · − +1 2−x −y 2−x −y x 2 + y 2 + xy − 2y − 2x + 1 = max 0, − . (5) 2−x −y Then from expression x 2 + y 2 + xy − 2y − 2x + 1 =0 2−x −y we obtain the formula for the margin of the considered zero set in the next form: y 2 + (x − 2)y + x 2 − 2x + 1 = 0. The only proper solution is: 3 x h(x) = 1 − − x − x 2 . 2 4 Its second derivative is given by: 1 − d 2 h(x) 3 = > 0, ∀x ∈]0, 1[. dx 2 3 2 4 x x− x x− 3 4
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Fig. 1. Pickands dependence function D and Archimax copula Cg,D .
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Fig. 2. Pickands dependence function D.
Because the second derivative is positive on ]0, 1[, the function h is convex on I and it really describes the zero set ZCg,D . To show that Cg,D is a conic copula, we apply a parametrization y = ax + b, a > 0. From the fact that copula Cg,D has to be linear on segments containing the point (1, 1), we have ax + b = 1, x = 1 and from this we have b = 1 − a. From (5) we see the piecewise linearity of copula Cg,D on the corresponding segment, namely (a 2 + a + 1)x 2 − (2a 2 + a + 1)x + a 2 (a 2 + a + 1)x − a 2 Cg,D (x, y) = max 0, = max 0, . (x − 1)(a + 1) a+1 Consequently, copula Cg,D is linear on the maximal segment of the straight line y = cx + 1 − c (where c = 1 a + a+1 ), which is a subset of I 2 . We have demonstrated the linearity of the copula Cg,D on linear segments connecting the points of the zero set margin with the point (1, 1, 0) and convexity of the function h describing the zero set margin, i.e., Cg,D = C[h] . In the following example we show the transition from the class of conic copulas to the class of Archimax copulas for chosen conic copula.
Example 3.2. Let h : 0, 12 → I be given by h(x) = (1 − 2x)2 . Obviously, h satisfies all constraints of Theorem 2.3. Then the conic copula C[h] (Figs. 2, 3) is defined as follows [8]: ⎧ 0 0 ≤ x ≤ 12 , y ≤ (1 − 2x)2 , ⎪ ⎪ ⎪ ⎪ ⎨ C[h] (x, y) = 4x(1 − x) − 1 + y (1 − 2x)2 < y and y ≥ 2x − 1, ⎪ 4(1 − x) − 1 + y ⎪ ⎪ ⎪ ⎩ min(x, y) else. Our aim is to represent the conic copula C[h] in the Archimax form Cg,D , see formula (3). 1−x For (x, y) ∈]0, 1[2 , z = , then for a chosen constant value z ∈ ]0, 1[ it holds: 2−x −y y=
1−z 2z − 1 x+ , z z
z = 0.
From (3) we obtain the formula D(z) =
(6) 1 − C[h] (x, y) describing a dependence function D. 2−x −y
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Fig. 3. Conic copula C[h] .
From (3) and (6) we get:
D(z) =
⎧ ⎪ ⎪ 1−z ⎪ ⎨ ⎪ ⎪ 4z2 ⎪ ⎩ 5z − 1
1 , z ∈ 0, 3 1 z ∈ ,1 . 3
The function D(z) is continuous and we show its convexity. 1 1 The function D1 = D| 0, is given by D1 (z) = 1 − z, z ∈ 0, . It is linear and therefore convex, and 3 3 − 1 dD1 3 = −1. dz 1 4z2 8 d 2 D2 (z) The function D2 = D| , 1 , D2 (z) = = > 0, and thus D2 is is twice differentiable, 2 3 5z − 1 dz (5z − 1)3 + 1 dD2 3 = −1, therefore the function D is convex on I and fulfills all conditions of Pickands convex. Moreover dz dependence function in definition of Archimax copulas. 4. Some other remarks Our previous results have an interesting consequence that Archimax copulas are isomorphic transforms of EV -copulas and of conic copulas. Observe that due to the isomorphism φ : I → I of a fixed (but arbitrarily chosen) non-strict Archimedean copula C and W (i.e., φ is a concave automorphism and f = 1 − φ = g ◦ φ is an additive generator of C), for the related Archimax copulas it holds Cf,D (x, y) = f =φ
−1
(−1)
(f (x) + f (y)) · D
f (x) f (x) + f (y)
1 − φ(x) max (0, (2 − φ(x) − φ(y)) · D 2 − φ(x) − φ(y)
,
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Fig. 4. Dependence functions D1 and D2 .
i.e., φ is an isomorphism between Archimax copulas Cf,D and Cg,D . Therefore, any Archimax copula Cf,D based on an additive generator f of a non-strict Archimedean copula C can be seen as a φ-transform of a conic copula Cf,D (x, y) = φ −1 C[h] (φ(x), φ(y)) . A similar relation is valid for the case of strict copulas. In fact, they are isomorphic to the product copula, and hence each Archimax copula Cf,D , such that Cf is a strict copula, is isomorphic to an extreme value copula linked to the Pickands dependence function D. Summarizing, we have the next observation: a copula C is an Archimax copula if and only if it is either isomorphic to some EV -copula [10], or to some conic copula. Another interesting result concerns the relation between conic quasi-copulas further studied in [8] and Archimax quasi-copulas, which can be defined similarly to Archimax copulas as follows. Definition 4.1. Let f : I → R be a function and let z0 ∈ I be fixed. We say that f is convex in point z0 if for any y, λ ∈ I it holds f (λz0 + (1 − λ) y) ≤ λf (z0 ) + (1 − λ) f (y) . Definition 4.2. Let f be an additive generator of some Archimedean copula and let D : I → I be a function convex in points z1 = 0 and z2 = 1, bounded from below by max (z, 1 − z). Then the function Qf,D : I 2 → I given by f (x) Qf,D (x, y) = f (−1) (f (x) + f (y)) · D f (x) + f (y) is called an Archimax quasi-copula. Example 4.1. Let D1 (z) = max −z + 1, 3z + 23 and D2 (z) = max − 3z + 1, z be two Pickands dependence functions, see Fig. 4. Then the function D (z) = min (D1 (z) , D2 (z)) is obviously not convex, but it is convex in points 0 and 1. 1−x It is easy to see that the function Qg,D given by Qg,D (x, y) = 1 − (2 − x − y) · D 2−x−y is a quasi-copula but not a copula. Indeed Qg,D (0, 0) = 0, Q Qg,D (x,1) = x and (1, y)= y, but themargin of the Qg,D g,D (1,1) =1,
zero set is composed of line segments (0, 1) , 0, 23 , 0, 23 , 25 , 25 , 25 , 25 , 23 , 0 and 23 , 0 , (1, 0) . The margin of the zero set is not convex (see Fig. 5), therefore the function Qg,D (see Fig. 6) is not a copula, but it fulfills all conditions of quasi-copula, i.e., Qg,D is a proper quasi-copula.
Because of [11] we know that supremum of every non-empty set of copulas is a quasi-copula. Moreover, any function D characterized in Definition 4.2 can be obtained as an infimum of Pickands dependence functions, D =
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Fig. 5. Dependence function D and margin of zero set ZQg,D .
Fig. 6. Quasi-copula Qg,D .
inf{Dt | t ∈ T } and thus any W -based Archimax quasi-copula Qg,D is a quasi-copula, Qg,D = sup{Cg,Dt | t ∈ T } = sup{C[ht ] | t ∈ T } = Q[ht ] , where ht = inf{ht | t ∈ T }. For the definition of conic quasi-copulas Q[h] see [7,8]. 5. Conclusion We have shown the equality of classes of conic copulas and of Archimax copulas related to the weakest copula W , including a similar result for quasi-copulas. This class ranges from W to M and thus we expect its applications in fitting copulas to real world problems, where there is no a priori known constraint on dependence parameters, such as Kendall’s τ or Spearman’s ρ. Moreover, we have stressed the characterization of the class of all Archimax copulas as isomorphic transforms of either EV -copulas or conic copulas. To stress the impact of our result, note that the computation of Spearman’s ρ for a W -based Archimax copula Cg,D given by (3) can be rather complicated. However, if we express Cg,D as a conic copula, Cg,D = C[h] , then it is not difficult to check that
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ρCg,D = ρC[h]
⎞ ⎛ d d 1 ⎝ ⎠ = 12 · · 1 − h(x) dx − 3 = 1 − 4 h(x) dx. 3 0
0
Acknowledgement This work was supported by grants VEGA 1/0420/15 and APVV 14-0013. The authors express their gratitude to anonymous referees for all comments which have helped them in improving the original version of this paper. References [1] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Stud. Fuzziness Soft Comput., vol. 221, Springer, Berlin, 2007. [2] P. Capéraà, A.-L. Fougères, C. Genest, Bivariate distributions with given extreme value attractor, J. Multivar. Anal. 72 (2000) 30–49. [3] F. Durante, P. Jaworski, Invariant dependence structure under univariate truncation, Statistics 46 (2) (2012) 263–277. [4] F. Durante, C. Sempi, Principles of Copula Theory, Chapman & Hall, London, 2015. [5] J. Fernandéz-Sanchéz, M. Úbeda-Flores, The distribution of the probability mass of conic copulas, Fuzzy Sets Syst. 284 (2016) 138–145. [6] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Encycl. Math. Appl., vol. 127, Cambridge University Press, 2009. [7] T. Jwaid, Semilinear and semiquadratic conjunctive aggregation function, PhD Thesis, Faculty of Sciences, Ghent University, Ghent, Belgium, 2014. [8] T. Jwaid, B. De Baets, J. Kalická, R. Mesiar, Conic aggregation functions, Fuzzy Sets Syst. 167 (2011) 3–20. [9] T. Jwaid, B. De Baets, H. De Meyer, The role of generalized convexity in conic copula constructions, J. Math. Anal. Appl. 425 (2015) 864–885. [10] R. Mesiar, V. Jágr, d-dimensional dependence functions and archimax copulas, Fuzzy Sets Syst. 228 (2013) 78–87. [11] R.B. Nelsen, An Introduction to Copulas, Springer, NY, ISBN 978-0-387-28659-4, 2006.