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1-Lipschitz power stable aggregation functions
Information Sciences 294 (2015) 57–63
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1-Lipschitz power stable aggregation functions Anna Kolesárová a,⇑, Radko Mesiar b,c a Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinského 9, 812 37 Bratislava 1, Slovakia b Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 813 68 Bratislava 1, Slovakia c University of Ostrava, Institute for Research and Applications of Fuzzy Modelling, 30. dubna 22, Ostrava, Czech Republic
a r t i c l e
i n f o
Article history: Received 11 April 2013 Received in revised form 27 May 2014 Accepted 17 September 2014 Available online 5 October 2014 Keywords: Aggregation function Copula Dependence function Power stability Quasi-copula
a b s t r a c t In this paper we study the structure of the class of all binary 1-Lipschitz power stable aggregation functions. A special attention is devoted to the characterization of power stable quasi-copulas. Moreover, the relations between copulas and quasi-copulas as 1-Lipschitz aggregation functions with neutral element e ¼ 1, sup- and inf-closures of copulas and ‘‘track-defined’’ functions based on copulas in the framework of binary power stable aggregation functions are characterized. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction General binary power stable aggregation functions have recently been characterized in [10]. Till now, no attempt has been made to characterize 1-Lipschitz power stable aggregation functions. One of the aims of this paper is to clarify the structure of the class of all binary 1-Lipschitz power stable aggregation functions. Our main attention will be devoted to the investigation of the power stability of special subclasses of 1-Lipschitz binary aggregation functions, namely copulas [11,14] and quasi-copulas [1,5], and also to the study of their relationships. Power stable copulas, which are also known as Extreme Value copulas (EV-copulas for short), were introduced, studied and shown to be of great importance in extreme value statistics dealing with financial, hydrological, insurance, and other problems [2,13]. For an exhaustive overview of EVcopulas we refer to [9]. Generalizations of copulas based on the coincidence of values on tracks are known as quasi-copulas [1]. They were also shown to be 1-Lipschitz aggregation functions with neutral element e ¼ 1, see [5]. It is also known [12] that the class of all quasi-copulas is the lattice closure of the class of all copulas. Even more, for each quasi-copula Q there exist systems of copulas ðC i Þi2I and ðDj Þj2J , such that Q ¼ supi2I C i ¼ inf j2J Dj (the standard partial ordering of 2-dimensional real functions is considered). The aim of this paper is to study the relations between copulas and quasi-copulas as 1-Lipschitz aggregation functions with neutral element e ¼ 1, sup- and inf-closures of copulas and ‘‘track-defined’’ functions based on copulas in the framework of binary power stable aggregation functions. The paper is organized as follows. In the next section, we introduce basic notions and results concerning binary power stable aggregation functions and EV-copulas. Section 3 is devoted to the study of binary 1-Lipschitz power stable aggregation ⇑ Corresponding author. E-mail addresses: [email protected] (A. Kolesárová), [email protected] (R. Mesiar). http://dx.doi.org/10.1016/j.ins.2014.09.034 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
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functions, and in Section 4 we clarify the relation between EV-copulas and power stable quasi-copulas. Finally, some concluding remarks are added. 2. EV-copulas and power stable aggregation functions We start by recalling several necessary notions and definitions. A function A : ½0; 12 ! ½0; 1 is called an aggregation function if it is increasing in each variable and satisfies the boundary conditions Að0; 0Þ ¼ 0 and Að1; 1Þ ¼ 1. For more information on aggregation functions we recommend the monographs [3,6] and overview papers [7,8]. Copulas and quasi-copulas are special aggregation functions. A copula is an aggregation function C : ½0; 12 ! ½0; 1 with neutral element e ¼ 1 satisfying the supermodular property, i.e., for all x; y 2 ½0; 12 ,
Cðx ^ yÞ þ Cðx _ yÞ P CðxÞ þ CðyÞ: A quasi-copula is an aggregation function Q : ½0; 12 ! ½0; 1 with neutral element e ¼ 1 that is 1-Lipschitz wrt. L1 -norm, i.e., for all x ¼ ðx1 ; y1 Þ; y ¼ ðx2 ; y2 Þ in ½0; 12 ,
jQ ðxÞ Q ðyÞj 6 jjx yjj1 ¼ jx1 x2 j þ jy1 y2 j: For more details concerning copulas and quasi-copulas we recommend the lecture notes [11] and a special issue of Information Sciences devoted to copulas, measures and integrals, see [4]. Definition 1. An aggregation function A : ½0; 12 ! ½0; 1 is called power stable whenever for any constant r 2 0; 1½ and all ðx; yÞ 2 ½0; 12 it holds
Aðxr ; yr Þ ¼ ðAðx; yÞÞr :
ð1Þ
The notion of power stable copulas is closely related to the notion of EV-copulas. In [2,11], EV-copulas were characterized as follows. Theorem 1. A copula C : ½0; 12 ! ½0; 1 is an EV-copula if and only if there is a convex function d : ½0; 1 ! ½0; 1 such that for all t 2 ½0; 1,
maxft; 1 tg 6 dðtÞ 6 1;
ð2Þ
2
and for all ðx; yÞ 2 0; 1½ , d
log x log xy
Cðx; yÞ ¼ ðxyÞ
:
ð3Þ
Remark 1. In fact the classes of all power stable copulas and all EV-copulas coincide. Recall that the EV-copulas are copulas preserving the stochastic dependence structure of random variables ðX; YÞ under maxima, i.e., for each fixed positive integer k, if random vectors ðX 1 ; X 2 Þ; . . . ; ðX k ; Y k Þ are independent and identically distributed (i.e., the same copula C corresponds to each couple ðX i ; Y i Þ), then the random vector ðmaxðX 1 ; . . . ; X k Þ; maxðY 1 ; . . . ; Y k ÞÞ is also characterized by the copula C. This means that C is stable with respect to the powers 1=k; k ¼ 1; 2; . . .. Due to Theorem 1, see [2,11], each such copula is also power stable, that is stable with respect to any power r in 0; 1½. The opposite inclusion is trivial. General continuous binary power stable aggregation functions have been characterized in our recent paper [10]. Theorem 2. A continuous function A : ½0; 12 ! ½0; 1 is a power stable aggregation function if and only if there is a continuous non-zero function d : ½0; 1 ! ½0; 1½ such that (i) the function dþ : 0; 1 ! ½0; 1½, dþ ðtÞ ¼ dðtÞ , is decreasing, and dð0Þ > 0 or limt!0þ dþ ðtÞ < 1, t dðtÞ (ii) the function d : ½0; 1½ ! ½0; 1½, d ðtÞ ¼ 1t , is increasing, and dð1Þ > 0 or limt!1 d ðtÞ < 1, and for all ðx; yÞ 2 0; 1½2 it holds d
Aðx; yÞ ¼ ðxyÞ
log x log xy
:
ð4Þ
The functions d characterized in Theorem 2 are called dependence functions, and the class of all such functions will be denoted by D. We will say that A is generated by a dependence function d. The dependence functions connected with EVcopulas, see Theorem 1, are known as the Pickands dependence functions, and the class of all Pickands’ dependence functions will be denoted by P. Note that each power stable aggregation function A is continuous on 0; 1½2 . Due to power stability, Aj0;1½2 0 or Aj0;1½2 1, or A is on 0; 1½2 generated by a function d via (4). If the function d satisfies all conditions given in Theorem 2, then there exists a continuous power stable aggregation function B : ½0; 12 ! ½0; 1 such that Bj0;1½2 ¼ Aj0;1½2 . For more details we refer to [10].
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The following result will also be important for our next considerations. Its proof is given in [10]. Proposition 1. A continuous power stable aggregation function A generated by a dependence function d 2 D has a neutral element e ¼ 1 if and only if dð0Þ ¼ dð1Þ ¼ 1. 3. 1-Lipschitz power stable aggregation functions It can easily be checked that for any continuous power stable aggregation function A : ½0; 12 ! ½0; 1, the partial functions Að0; :Þ; Að:; 0Þ; Að1; :Þ and Að:; 1Þ are either constants 0 or 1, or they are power functions t a with a positive a. If A is considered to be 1-Lipschitz, then for the above partial functions there are only the following possibilities: Að0; :Þ; Að:; 0Þ 2 f0; idj½0;1 g and Að1; :Þ; Að:; 1Þ 2 f1; idj½0;1 g. Thus, using the monotonicity of aggregation functions, we can divide all 1-Lipschitz power stable aggregation functions into the following four classes. Namely, the class of aggregation functions for which (i) Að0; tÞ ¼ Aðt; 0Þ ¼ 0 (and then Að1; tÞ ¼ Aðt; 1Þ ¼ t), i.e., 1-Lipschitz power stable aggregation functions with neutral element e ¼ 1, which means that the first class is formed by power stable quasi-copulas. (ii) Að0; tÞ ¼ Að1; tÞ ¼ t; t 2 ½0; 1, which gives Aðx; yÞ ¼ y, i.e., the second class only contains A ¼ PL , the projection to the second coordinate. (iii) Aðt; 0Þ ¼ Aðt; 1Þ ¼ t; t 2 ½0; 1, which gives Aðx; yÞ ¼ x, i.e., the third class only contains A ¼ PF , the projection to the first coordinate. (iv) Að0; tÞ ¼ Aðt; 0Þ ¼ t (and then Að1; tÞ ¼ Aðt; 1Þ ¼ 1), t 2 ½0; 1. Then Aðt; tÞ ¼ t a ; a > 0 and due to the 1-Lipschitz property of A, for each t 2 ½0; 1 it holds
Að1; tÞ Aðt; tÞ ¼ 1 t a 6 1 t; i.e., a 2 0; 1. On the other hand, for each t 2 ½0; 1 it has to hold
Aðt; tÞ Aðt; 0Þ ¼ t a t 6 t; which implies a ¼ 1, and thus Aðt; tÞ ¼ t. Consequently, Aðx; yÞ ¼ maxfx; yg. The only non-trivial case which has to be clarified, is case (i) of the previous discussion concerning power stable quasicopulas. Recall that each quasi-copula Q is a conjunctive aggregation function, i.e., Q 6 Min, see, e.g., [6]. Proposition 2. A continuous power stable aggregation function A generated by a dependence function d is conjunctive if and only if for each t 2 ½0; 1,
dðtÞ P maxft; 1 tg: Proof. A is a conjunctive aggregation function if and only if A 6 Min, i.e., if for all ðx; yÞ 2 ½0; 12 ; Aðx; yÞ 6 minfx; yg. Observe that the inequality Aðx; yÞ 6 x is, for each ðx; yÞ 2 0; 1½2 , equivalent to
log x log x log x 6 exp log x and next to d P exp ðlog xyÞd ; log xy log xy log xy which can be written as dðtÞ P t; t 2 ½0; 1. On the other hand, the inequality Aðx; yÞ 6 y is equivalent to dðtÞ P 1 t. h Before giving a complete characterization of power stable quasi-copulas, recall that a real function u : ½a; b ! R is convex at a point t 0 2 ½a; b, if for each t 2 ½a; b and c 2 ½0; 1 it holds
f ðð1 cÞt0 þ ctÞ 6 ð1 cÞf ðt 0 Þ þ cf ðtÞ: Observe that u is convex on ½a; b if and only if it is convex at each point t0 2 ½a; b. In the next characterization we will use the convexity of a dependence function d 2 D at points 0 and 1, i.e., the properties
dðctÞ 6 ð1 cÞdð0Þ þ cdðtÞ;
ð5Þ
dðct þ 1 cÞ 6 ð1 cÞdð1Þ þ cdðtÞ;
ð6Þ
and
respectively, valid for all t 2 ½0; 1 and c 2 ½0; 1. Theorem 3. A continuous function Q : ½0; 12 ! ½0; 1 is a power stable quasi-copula if and only if it can be written in the form (4) for some dependence function d 2 D that is convex at the points 0 and 1 and dð0Þ ¼ dð1Þ ¼ 1.
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Proof. Sufficiency. First observe that for a dependence function d 2 D such that dð0Þ ¼ dð1Þ ¼ 1, the corresponding power stable aggregation function is conjunctive and thus, by Proposition 2, for all t 2 ½0; 1; maxft; 1 tg 6 dðtÞ. Next, for each u 20; 1½, consider the function du : ½0; 1 ! ½0; 1, defined by
( du ðtÞ ¼
1 ut þ ut dðuÞ if t 6 u; 1t tu dðuÞ þ 1u otherwise: 1u
Due to (5) and (6), du is a convex function satisfying all requirements for Pickands’ dependence functions, i.e., du 2 P, and thus it generates via (3) a copula C u . Moreover, d ¼ inf u20;1½ du and thus, Q ¼ supu20;1½ C u , i.e., Q is a quasi-copula. Necessity. Observe that the property dð0Þ ¼ dð1Þ ¼ 1 follows from Proposition 1. Suppose, e.g., that d is not 0-convex, i.e., there are t; c 20; 1½ such that dðctÞ > 1 c þ cdðtÞ. From the representation formula (4) we can see that for any x; y 2 ½0; 1½ such that
log x log xy
¼ t, i.e., y ¼ xð1tÞ=t , it holds dðtÞ
Qðx; yÞ ¼ Qðx; xð1tÞ=t Þ ¼ ðxxð1tÞ=t Þ
¼ xdðtÞ=t :
Similarly, Q ðx; xð1ctÞ=ct Þ ¼ xdðctÞ=ct . The 1-Lipschitz property of Q ensures that for all x 20; 1½,
Qðx; xð1tÞ=t Þ Qðx; xð1ctÞ=ct Þ ¼ xdðtÞ=t xdðctÞ=ct 6 xð1tÞ=t xð1ctÞ=ct : Next, the continuity of Q implies that the function f : 0; 1 ! R,
f ðxÞ ¼ xdðtÞ=t xdðctÞ=ct xð1tÞ=t þ xð1ctÞ=ct ; is for each x 20; 1 non-positive. On the other hand, it holds f ð1Þ ¼ 0 and 0
f ð1Þ ¼
dðtÞ dðctÞ 1 t 1 ct dðtÞ 1 c þ cdðtÞ 1 1 þ < þ ¼ 0; t ct t ct t ct t ct
i.e., f is strictly decreasing on some neighbourhood of the point 1. However, then necessarily f ðxÞ > 0 for some x close to 1, which is a contradiction. h Example 1. Consider the Pickands dependence functions d1 ; d2 2 P, given by
1þt ; d1 ðtÞ ¼ max 1 t; 2
d2 ðtÞ ¼ d1 ð1 tÞ:
pffiffiffi e, C e ðx; yÞ ¼ Cðy; xÞ. The function Then d1 generates the copula C given by Cðx; yÞ ¼ minfy; x yg, and d2 generates the copula C d ¼ minfd1 ; d2 g 2 D is a dependence function satisfying properties given in Theorem 3, and generates a proper power stable quasi-copula Q (i.e., Q is not a copula) given by Q ðx; yÞ ¼ maxfCðx; yÞ; Cðy; xÞg. To see that Q is not a copula, it is enough to
consider points x ¼ 14 ; 12 and y ¼ 12 ; 14 . Then
Qðx _ yÞ þ Q ðx ^ yÞ ¼ Q
pffiffiffi 1 1 1 1 2 2þ1 1 þQ ¼ ; ; < ¼ Q ðxÞ þ Q ðyÞ; 8 2 2 4 4 2
which means that Q is not supermodular (see Fig. 1).
Fig. 1. The function d 2 D, Example 1, convex at points 0 and 1, dð0Þ ¼ dð1Þ ¼ 1.
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Fig. 2. The dependence function d from Example 2 that is not convex at point 1.
Fig. 3. The aggregation function A generated by the dependence function d from Example 2.
Example 2. Consider the function d defined by
8 1t > > > < 1=3 þ t dðtÞ ¼ > 1 t=3 > > : t
t 2 ½0; 1=3; t 21=3; 1=2; t 21=2; 3=4; t 23=4; 1;
see Fig. 2. It is easy to see that d 2 D; dð0Þ ¼ dð1Þ ¼ 1, and that d is 0-convex, but not 1-convex. If we construct the aggregation function A using (4), we obtain
8 y > > > > < 4 1=3 ðx yÞ Aðx; yÞ ¼ > > xð2=3Þ y > > : x
x 20; 1½; 0 < y 6 x2 ; x 20; 1½; x2 < y 6 x; pffiffiffi x 20; 1½; x < y 6 3 x; p ffiffi ffi x 20; 1½; 3 x < y < 1;
see Fig. 3. A is 1-Lipschitz in the second variable, i.e., Aðx; :Þ is 1-Lipschitz for each x, but A is not 1-Lipschitz in the first variable. Indeed, if we choose, e.g., the points u ¼ ð0:9; 0:81Þ and v ¼ ð0:81; 0:81Þ, then Að0:9; 0:81Þ ¼ 0:81, p ffiffiffiffiffiffiffiffiffiffiffiffi 3 Að0:81; 0:81Þ ¼ 0:910 ¼ 0:703842, and thus,
Að0:9; 0:81Þ Að0:81; 0:81Þ ¼ 0:106158 > 0:09 ¼ 0:9 0:81; which confirms that A is not 1-Lipschitz in the first variable.
4. Power stable quasi-copulas and EV-copulas As already mentioned, quasi-copulas can be linked to copulas by means of several equivalent ways. Denote by A the class of all binary aggregation functions. For any subclass B # A, put
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B_ ¼
( ) _ Ai jI – ;; 8i 2 I; Ai 2 B ; i2I
( ) ^ ^ B ¼ Ai jI – ;; 8i 2 I; Ai 2 B ; i2I
BL ¼ fA 2 BjA is 1-Lipschitz with neutral element e ¼ 1g; BT ¼ fA 2 Aj8 track T 2 T 9 B 2 B; 8ðx; yÞ 2 T; Aðx; yÞ ¼ Bðx; yÞg: Recall that a track T ¼ fðFðtÞ; GðtÞÞjt 2 ½0; 1g is a subset of ½0; 12 defined by means of two increasing surjections F; G : ½0; 1 ! ½0; 1. Both F and G can be seen as continuous distribution functions. T denotes the set of all possible tracks. For the class of all binary copulas C and the class of all binary quasi-copulas Q we have already mentioned some facts, which can be summarized as follows:
C – C _ ¼ C ^ ¼ CT ¼ AL ¼ Q: We can see here four equivalent definitions of binary quasi-copulas. It is trivial to see that Q_ ¼ Q^ ¼ QT ¼ QL ¼ Q. Now, consider the class C PS of all power stable copulas (i.e., EV-copulas), the class QPS of all power stable quasi-copulas, and the class APS of all power stable binary aggregation functions. From Proposition 1 we get C PS ¼ C ^PS . This is true due to the closedness of the class P of all Pickands’ dependence functions with respect to suprema. On the other hand, due to Theorem 2, each dependence function generating a power stable quasi-copula Q 2 QPS can be obtained as the infimum of the Pickands dependence functions, i.e., Q can be obtained as the supremum of some copulas. Hence
CPS $ C _PS ¼ QPS ¼ ALPS ; (the last equality is a matter of definition only). As a non-trivial problem, it remains to clarify the position of C TPS , i.e., the class of all aggregation functions coinciding on each track with some power stable copula. Theorem 4. It holds C TPS ¼ C PS . Proof. Obviously, C PS # C TPS . Hence, it is enough to show that each aggregation function A coinciding on each track T with some EV-copula, is also an EV-copula. Choose an arbitrary A 2 C TPS and consider a track T a ¼ fðt; ta Þjt 2 ½0; 1g, for an a 2 0; 1½. Then there is an EV-copula C a 2 C PS such that for all t 2 ½0; 1; Aðt; ta Þ ¼ C a ðt; ta Þ. Thus for each power k 2 0; 1½ it holds a
k
a
k
k
k
Aðt k ; ðt a Þ Þ ¼ Aðt k ; ðt k Þ Þ ¼ C a ðtk ; ðtk Þ Þ ¼ C a ðtk ; ðt a Þ Þ ¼ ðC a ðt; ta ÞÞ ¼ ðAðt; t a ÞÞ : For each ðx; yÞ 2 0; 1½2 there is an a 2 0; 1½ such that y ¼ xa , and hence
Aðxk ; yk Þ ¼ ðAðx; yÞÞk ; i.e., A 2 APS . Moreover, because of the continuity of copulas, A is also continuous. So, we can apply Proposition 2, namely the representation of A in the form (4) with a dependence function d 2 D. Now, consider the track T generated by functions F; G : ½0; 1 ! ½0; 1 given by
FðtÞ ¼ minf1; maxf0; 3t 1gg and GðtÞ ¼ t:
Then there is an EV-copula C generated by a Pickands’ dependence function d 2 P such that AðFðtÞ; GðtÞÞ ¼ CðFðtÞ; GðtÞÞ, i.e., d
ðFðtÞGðtÞÞ for all t 2
1
;2 3 3
log FðtÞ log FðtÞGðtÞ
d
¼ ðFðtÞGðtÞÞ
log FðtÞ log FðtÞGðtÞ
;
. Consequently, for all t 2 13 ; 23 it holds
logð3t 1Þ logð3t 1Þ ¼d : d log tð3t 1Þ log tð3t 1Þ
logð3t1Þ The function f : 13 ; 23 ! R given by f ðtÞ ¼ log is continuous and limt!1þ f ðtÞ ¼ 1, limt!2 f ðtÞ ¼ 0, i.e., for any u 2 0; 1½ tð3t1Þ 3 1 2 3 there is a t 2 3 ; 3 such that u ¼ f ðtÞ and thus for each u 2 0; 1½; dðuÞ ¼ d ðuÞ. The continuity of both d and d results in d ¼ d on ½0; 1, hence, A ¼ C, i.e., A is an EV-copula. h 5. Concluding remarks We have clarified the structure of the class of all binary 1-Lipschitz power stable aggregation functions and the prominent role of power stable quasi-copulas. As the main result we have shown the relationship of power stable quasi-copulas and dependence functions convex at points 0 and 1, satisfying the property dð0Þ ¼ dð1Þ ¼ 1. We have also clarified the
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63
relationship of EV-copulas and power stable quasi-copulas. As a by-product of the result given in Theorem 4 is the fact that any EV-copula is determined by any of its non-trivial horizontal or vertical sections ha ; v a , respectively, where ha ðxÞ ¼ Cðx; aÞ and v a ðxÞ ¼ Cða; xÞ; a 2 0; 1½. This is not the case of a diagonal section. Indeed, a function d : ½0; 1 ! ½0; 1 is a diagonal section of some EV-copula C if and only if dðxÞ ¼ xa ; a 2 ½1; 2. Up to the trivial cases a ¼ 1 (when C ¼ M) and a ¼ 2 (when C ¼ P), for each a 2 1; 2½ there are infinitely many EV-copulas C such that Cðx; xÞ ¼ xa . Really, any Pickands’
dependence function d 2 P such that d 12 ¼ a2 generates such an EV-copula C. In our further investigation, we open the problem of characterization of 1-Lipschitz power stable aggregation functions of higher dimensions. Acknowledgements The authors kindly acknowledge the support of the grant VEGA 1/0419/13 and the project of Science and Technology Assistance Agency under the contract no. APVV-0073-10. Moreover, the work of R. Mesiar on this paper was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project reg. no. CZ.1.05/1.1.00/ 02.0070. References [1] C. Alsina, R.B. Nelsen, B. Schweizer, On the characterization of a class of binary operations on distributions functions, Stat. Probab. Lett. 17 (1993) 85–89. [2] J. Beirlant, Y. Goegebeur, J. Segers, J. Teugels, Statistics of Extremes: Theory and Applications, John Wiley & Sons Ltd., Chichester, 2004. [3] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer, Heidelberg, Berlin, New York, 2007. [4] F. Durante, R. Mesiar, S. Saminger-Platz, Inform. Sci. 179 (17) (2009). [5] C. Genest, J.J. Quesada Molina, J.A. Rodríguez-Lallena, C. Sempi, A characterization of quasi-copulas, J. Multivariate Anal. 69 (1999) 193–205. [6] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, Cambridge, 2009. [7] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation functions: means, Inform. Sci. 181 (2011) 1–22. [8] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation functions: construction methods, conjunctive, disjunctive and mixed classes, Inform. Sci. 181 (2011) 23–43. [9] G. Gudendorf, J. Segers, Extreme-value copulas, in: F. Durante et al. (Eds.), Proc. of Workshop on Copulas Theory and its Applications, 2010. [10] A. Kolesárová, R. Mesiar, T. Rückschlossová, Power stable aggregation functions, Fuzzy Sets Syst. 240 (2014) 39–50. [11] R.B. Nelsen, An introduction to copulas, Lecture Notes in Statistics, second ed., vol. 139, Springer, New York, 2006. [12] R.B. Nelsen, M. Úbeda-Flores, The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas, C.R. Acad. Sci. Paris Ser. I 341 (2005) 583–586. [13] G. Salvadori, C. De Michele, N.T. Kottegoda, R. Rosso, Extremes in Nature, An Approach Using Copulas, Springer, Dordrecht, 2007. [14] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231.
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1-Lipschitz power stable aggregation functions Anna Kolesárová a,⇑, Radko Mesiar b,c a Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology in Bratislava, Radlinského 9, 812 37 Bratislava 1, Slovakia b Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 813 68 Bratislava 1, Slovakia c University of Ostrava, Institute for Research and Applications of Fuzzy Modelling, 30. dubna 22, Ostrava, Czech Republic
a r t i c l e
i n f o
Article history: Received 11 April 2013 Received in revised form 27 May 2014 Accepted 17 September 2014 Available online 5 October 2014 Keywords: Aggregation function Copula Dependence function Power stability Quasi-copula
a b s t r a c t In this paper we study the structure of the class of all binary 1-Lipschitz power stable aggregation functions. A special attention is devoted to the characterization of power stable quasi-copulas. Moreover, the relations between copulas and quasi-copulas as 1-Lipschitz aggregation functions with neutral element e ¼ 1, sup- and inf-closures of copulas and ‘‘track-defined’’ functions based on copulas in the framework of binary power stable aggregation functions are characterized. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction General binary power stable aggregation functions have recently been characterized in [10]. Till now, no attempt has been made to characterize 1-Lipschitz power stable aggregation functions. One of the aims of this paper is to clarify the structure of the class of all binary 1-Lipschitz power stable aggregation functions. Our main attention will be devoted to the investigation of the power stability of special subclasses of 1-Lipschitz binary aggregation functions, namely copulas [11,14] and quasi-copulas [1,5], and also to the study of their relationships. Power stable copulas, which are also known as Extreme Value copulas (EV-copulas for short), were introduced, studied and shown to be of great importance in extreme value statistics dealing with financial, hydrological, insurance, and other problems [2,13]. For an exhaustive overview of EVcopulas we refer to [9]. Generalizations of copulas based on the coincidence of values on tracks are known as quasi-copulas [1]. They were also shown to be 1-Lipschitz aggregation functions with neutral element e ¼ 1, see [5]. It is also known [12] that the class of all quasi-copulas is the lattice closure of the class of all copulas. Even more, for each quasi-copula Q there exist systems of copulas ðC i Þi2I and ðDj Þj2J , such that Q ¼ supi2I C i ¼ inf j2J Dj (the standard partial ordering of 2-dimensional real functions is considered). The aim of this paper is to study the relations between copulas and quasi-copulas as 1-Lipschitz aggregation functions with neutral element e ¼ 1, sup- and inf-closures of copulas and ‘‘track-defined’’ functions based on copulas in the framework of binary power stable aggregation functions. The paper is organized as follows. In the next section, we introduce basic notions and results concerning binary power stable aggregation functions and EV-copulas. Section 3 is devoted to the study of binary 1-Lipschitz power stable aggregation ⇑ Corresponding author. E-mail addresses: [email protected] (A. Kolesárová), [email protected] (R. Mesiar). http://dx.doi.org/10.1016/j.ins.2014.09.034 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
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functions, and in Section 4 we clarify the relation between EV-copulas and power stable quasi-copulas. Finally, some concluding remarks are added. 2. EV-copulas and power stable aggregation functions We start by recalling several necessary notions and definitions. A function A : ½0; 12 ! ½0; 1 is called an aggregation function if it is increasing in each variable and satisfies the boundary conditions Að0; 0Þ ¼ 0 and Að1; 1Þ ¼ 1. For more information on aggregation functions we recommend the monographs [3,6] and overview papers [7,8]. Copulas and quasi-copulas are special aggregation functions. A copula is an aggregation function C : ½0; 12 ! ½0; 1 with neutral element e ¼ 1 satisfying the supermodular property, i.e., for all x; y 2 ½0; 12 ,
Cðx ^ yÞ þ Cðx _ yÞ P CðxÞ þ CðyÞ: A quasi-copula is an aggregation function Q : ½0; 12 ! ½0; 1 with neutral element e ¼ 1 that is 1-Lipschitz wrt. L1 -norm, i.e., for all x ¼ ðx1 ; y1 Þ; y ¼ ðx2 ; y2 Þ in ½0; 12 ,
jQ ðxÞ Q ðyÞj 6 jjx yjj1 ¼ jx1 x2 j þ jy1 y2 j: For more details concerning copulas and quasi-copulas we recommend the lecture notes [11] and a special issue of Information Sciences devoted to copulas, measures and integrals, see [4]. Definition 1. An aggregation function A : ½0; 12 ! ½0; 1 is called power stable whenever for any constant r 2 0; 1½ and all ðx; yÞ 2 ½0; 12 it holds
Aðxr ; yr Þ ¼ ðAðx; yÞÞr :
ð1Þ
The notion of power stable copulas is closely related to the notion of EV-copulas. In [2,11], EV-copulas were characterized as follows. Theorem 1. A copula C : ½0; 12 ! ½0; 1 is an EV-copula if and only if there is a convex function d : ½0; 1 ! ½0; 1 such that for all t 2 ½0; 1,
maxft; 1 tg 6 dðtÞ 6 1;
ð2Þ
2
and for all ðx; yÞ 2 0; 1½ , d
log x log xy
Cðx; yÞ ¼ ðxyÞ
:
ð3Þ
Remark 1. In fact the classes of all power stable copulas and all EV-copulas coincide. Recall that the EV-copulas are copulas preserving the stochastic dependence structure of random variables ðX; YÞ under maxima, i.e., for each fixed positive integer k, if random vectors ðX 1 ; X 2 Þ; . . . ; ðX k ; Y k Þ are independent and identically distributed (i.e., the same copula C corresponds to each couple ðX i ; Y i Þ), then the random vector ðmaxðX 1 ; . . . ; X k Þ; maxðY 1 ; . . . ; Y k ÞÞ is also characterized by the copula C. This means that C is stable with respect to the powers 1=k; k ¼ 1; 2; . . .. Due to Theorem 1, see [2,11], each such copula is also power stable, that is stable with respect to any power r in 0; 1½. The opposite inclusion is trivial. General continuous binary power stable aggregation functions have been characterized in our recent paper [10]. Theorem 2. A continuous function A : ½0; 12 ! ½0; 1 is a power stable aggregation function if and only if there is a continuous non-zero function d : ½0; 1 ! ½0; 1½ such that (i) the function dþ : 0; 1 ! ½0; 1½, dþ ðtÞ ¼ dðtÞ , is decreasing, and dð0Þ > 0 or limt!0þ dþ ðtÞ < 1, t dðtÞ (ii) the function d : ½0; 1½ ! ½0; 1½, d ðtÞ ¼ 1t , is increasing, and dð1Þ > 0 or limt!1 d ðtÞ < 1, and for all ðx; yÞ 2 0; 1½2 it holds d
Aðx; yÞ ¼ ðxyÞ
log x log xy
:
ð4Þ
The functions d characterized in Theorem 2 are called dependence functions, and the class of all such functions will be denoted by D. We will say that A is generated by a dependence function d. The dependence functions connected with EVcopulas, see Theorem 1, are known as the Pickands dependence functions, and the class of all Pickands’ dependence functions will be denoted by P. Note that each power stable aggregation function A is continuous on 0; 1½2 . Due to power stability, Aj0;1½2 0 or Aj0;1½2 1, or A is on 0; 1½2 generated by a function d via (4). If the function d satisfies all conditions given in Theorem 2, then there exists a continuous power stable aggregation function B : ½0; 12 ! ½0; 1 such that Bj0;1½2 ¼ Aj0;1½2 . For more details we refer to [10].
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The following result will also be important for our next considerations. Its proof is given in [10]. Proposition 1. A continuous power stable aggregation function A generated by a dependence function d 2 D has a neutral element e ¼ 1 if and only if dð0Þ ¼ dð1Þ ¼ 1. 3. 1-Lipschitz power stable aggregation functions It can easily be checked that for any continuous power stable aggregation function A : ½0; 12 ! ½0; 1, the partial functions Að0; :Þ; Að:; 0Þ; Að1; :Þ and Að:; 1Þ are either constants 0 or 1, or they are power functions t a with a positive a. If A is considered to be 1-Lipschitz, then for the above partial functions there are only the following possibilities: Að0; :Þ; Að:; 0Þ 2 f0; idj½0;1 g and Að1; :Þ; Að:; 1Þ 2 f1; idj½0;1 g. Thus, using the monotonicity of aggregation functions, we can divide all 1-Lipschitz power stable aggregation functions into the following four classes. Namely, the class of aggregation functions for which (i) Að0; tÞ ¼ Aðt; 0Þ ¼ 0 (and then Að1; tÞ ¼ Aðt; 1Þ ¼ t), i.e., 1-Lipschitz power stable aggregation functions with neutral element e ¼ 1, which means that the first class is formed by power stable quasi-copulas. (ii) Að0; tÞ ¼ Að1; tÞ ¼ t; t 2 ½0; 1, which gives Aðx; yÞ ¼ y, i.e., the second class only contains A ¼ PL , the projection to the second coordinate. (iii) Aðt; 0Þ ¼ Aðt; 1Þ ¼ t; t 2 ½0; 1, which gives Aðx; yÞ ¼ x, i.e., the third class only contains A ¼ PF , the projection to the first coordinate. (iv) Að0; tÞ ¼ Aðt; 0Þ ¼ t (and then Að1; tÞ ¼ Aðt; 1Þ ¼ 1), t 2 ½0; 1. Then Aðt; tÞ ¼ t a ; a > 0 and due to the 1-Lipschitz property of A, for each t 2 ½0; 1 it holds
Að1; tÞ Aðt; tÞ ¼ 1 t a 6 1 t; i.e., a 2 0; 1. On the other hand, for each t 2 ½0; 1 it has to hold
Aðt; tÞ Aðt; 0Þ ¼ t a t 6 t; which implies a ¼ 1, and thus Aðt; tÞ ¼ t. Consequently, Aðx; yÞ ¼ maxfx; yg. The only non-trivial case which has to be clarified, is case (i) of the previous discussion concerning power stable quasicopulas. Recall that each quasi-copula Q is a conjunctive aggregation function, i.e., Q 6 Min, see, e.g., [6]. Proposition 2. A continuous power stable aggregation function A generated by a dependence function d is conjunctive if and only if for each t 2 ½0; 1,
dðtÞ P maxft; 1 tg: Proof. A is a conjunctive aggregation function if and only if A 6 Min, i.e., if for all ðx; yÞ 2 ½0; 12 ; Aðx; yÞ 6 minfx; yg. Observe that the inequality Aðx; yÞ 6 x is, for each ðx; yÞ 2 0; 1½2 , equivalent to
log x log x log x 6 exp log x and next to d P exp ðlog xyÞd ; log xy log xy log xy which can be written as dðtÞ P t; t 2 ½0; 1. On the other hand, the inequality Aðx; yÞ 6 y is equivalent to dðtÞ P 1 t. h Before giving a complete characterization of power stable quasi-copulas, recall that a real function u : ½a; b ! R is convex at a point t 0 2 ½a; b, if for each t 2 ½a; b and c 2 ½0; 1 it holds
f ðð1 cÞt0 þ ctÞ 6 ð1 cÞf ðt 0 Þ þ cf ðtÞ: Observe that u is convex on ½a; b if and only if it is convex at each point t0 2 ½a; b. In the next characterization we will use the convexity of a dependence function d 2 D at points 0 and 1, i.e., the properties
dðctÞ 6 ð1 cÞdð0Þ þ cdðtÞ;
ð5Þ
dðct þ 1 cÞ 6 ð1 cÞdð1Þ þ cdðtÞ;
ð6Þ
and
respectively, valid for all t 2 ½0; 1 and c 2 ½0; 1. Theorem 3. A continuous function Q : ½0; 12 ! ½0; 1 is a power stable quasi-copula if and only if it can be written in the form (4) for some dependence function d 2 D that is convex at the points 0 and 1 and dð0Þ ¼ dð1Þ ¼ 1.
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Proof. Sufficiency. First observe that for a dependence function d 2 D such that dð0Þ ¼ dð1Þ ¼ 1, the corresponding power stable aggregation function is conjunctive and thus, by Proposition 2, for all t 2 ½0; 1; maxft; 1 tg 6 dðtÞ. Next, for each u 20; 1½, consider the function du : ½0; 1 ! ½0; 1, defined by
( du ðtÞ ¼
1 ut þ ut dðuÞ if t 6 u; 1t tu dðuÞ þ 1u otherwise: 1u
Due to (5) and (6), du is a convex function satisfying all requirements for Pickands’ dependence functions, i.e., du 2 P, and thus it generates via (3) a copula C u . Moreover, d ¼ inf u20;1½ du and thus, Q ¼ supu20;1½ C u , i.e., Q is a quasi-copula. Necessity. Observe that the property dð0Þ ¼ dð1Þ ¼ 1 follows from Proposition 1. Suppose, e.g., that d is not 0-convex, i.e., there are t; c 20; 1½ such that dðctÞ > 1 c þ cdðtÞ. From the representation formula (4) we can see that for any x; y 2 ½0; 1½ such that
log x log xy
¼ t, i.e., y ¼ xð1tÞ=t , it holds dðtÞ
Qðx; yÞ ¼ Qðx; xð1tÞ=t Þ ¼ ðxxð1tÞ=t Þ
¼ xdðtÞ=t :
Similarly, Q ðx; xð1ctÞ=ct Þ ¼ xdðctÞ=ct . The 1-Lipschitz property of Q ensures that for all x 20; 1½,
Qðx; xð1tÞ=t Þ Qðx; xð1ctÞ=ct Þ ¼ xdðtÞ=t xdðctÞ=ct 6 xð1tÞ=t xð1ctÞ=ct : Next, the continuity of Q implies that the function f : 0; 1 ! R,
f ðxÞ ¼ xdðtÞ=t xdðctÞ=ct xð1tÞ=t þ xð1ctÞ=ct ; is for each x 20; 1 non-positive. On the other hand, it holds f ð1Þ ¼ 0 and 0
f ð1Þ ¼
dðtÞ dðctÞ 1 t 1 ct dðtÞ 1 c þ cdðtÞ 1 1 þ < þ ¼ 0; t ct t ct t ct t ct
i.e., f is strictly decreasing on some neighbourhood of the point 1. However, then necessarily f ðxÞ > 0 for some x close to 1, which is a contradiction. h Example 1. Consider the Pickands dependence functions d1 ; d2 2 P, given by
1þt ; d1 ðtÞ ¼ max 1 t; 2
d2 ðtÞ ¼ d1 ð1 tÞ:
pffiffiffi e, C e ðx; yÞ ¼ Cðy; xÞ. The function Then d1 generates the copula C given by Cðx; yÞ ¼ minfy; x yg, and d2 generates the copula C d ¼ minfd1 ; d2 g 2 D is a dependence function satisfying properties given in Theorem 3, and generates a proper power stable quasi-copula Q (i.e., Q is not a copula) given by Q ðx; yÞ ¼ maxfCðx; yÞ; Cðy; xÞg. To see that Q is not a copula, it is enough to
consider points x ¼ 14 ; 12 and y ¼ 12 ; 14 . Then
Qðx _ yÞ þ Q ðx ^ yÞ ¼ Q
pffiffiffi 1 1 1 1 2 2þ1 1 þQ ¼ ; ; < ¼ Q ðxÞ þ Q ðyÞ; 8 2 2 4 4 2
which means that Q is not supermodular (see Fig. 1).
Fig. 1. The function d 2 D, Example 1, convex at points 0 and 1, dð0Þ ¼ dð1Þ ¼ 1.
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Fig. 2. The dependence function d from Example 2 that is not convex at point 1.
Fig. 3. The aggregation function A generated by the dependence function d from Example 2.
Example 2. Consider the function d defined by
8 1t > > > < 1=3 þ t dðtÞ ¼ > 1 t=3 > > : t
t 2 ½0; 1=3; t 21=3; 1=2; t 21=2; 3=4; t 23=4; 1;
see Fig. 2. It is easy to see that d 2 D; dð0Þ ¼ dð1Þ ¼ 1, and that d is 0-convex, but not 1-convex. If we construct the aggregation function A using (4), we obtain
8 y > > > > < 4 1=3 ðx yÞ Aðx; yÞ ¼ > > xð2=3Þ y > > : x
x 20; 1½; 0 < y 6 x2 ; x 20; 1½; x2 < y 6 x; pffiffiffi x 20; 1½; x < y 6 3 x; p ffiffi ffi x 20; 1½; 3 x < y < 1;
see Fig. 3. A is 1-Lipschitz in the second variable, i.e., Aðx; :Þ is 1-Lipschitz for each x, but A is not 1-Lipschitz in the first variable. Indeed, if we choose, e.g., the points u ¼ ð0:9; 0:81Þ and v ¼ ð0:81; 0:81Þ, then Að0:9; 0:81Þ ¼ 0:81, p ffiffiffiffiffiffiffiffiffiffiffiffi 3 Að0:81; 0:81Þ ¼ 0:910 ¼ 0:703842, and thus,
Að0:9; 0:81Þ Að0:81; 0:81Þ ¼ 0:106158 > 0:09 ¼ 0:9 0:81; which confirms that A is not 1-Lipschitz in the first variable.
4. Power stable quasi-copulas and EV-copulas As already mentioned, quasi-copulas can be linked to copulas by means of several equivalent ways. Denote by A the class of all binary aggregation functions. For any subclass B # A, put
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B_ ¼
( ) _ Ai jI – ;; 8i 2 I; Ai 2 B ; i2I
( ) ^ ^ B ¼ Ai jI – ;; 8i 2 I; Ai 2 B ; i2I
BL ¼ fA 2 BjA is 1-Lipschitz with neutral element e ¼ 1g; BT ¼ fA 2 Aj8 track T 2 T 9 B 2 B; 8ðx; yÞ 2 T; Aðx; yÞ ¼ Bðx; yÞg: Recall that a track T ¼ fðFðtÞ; GðtÞÞjt 2 ½0; 1g is a subset of ½0; 12 defined by means of two increasing surjections F; G : ½0; 1 ! ½0; 1. Both F and G can be seen as continuous distribution functions. T denotes the set of all possible tracks. For the class of all binary copulas C and the class of all binary quasi-copulas Q we have already mentioned some facts, which can be summarized as follows:
C – C _ ¼ C ^ ¼ CT ¼ AL ¼ Q: We can see here four equivalent definitions of binary quasi-copulas. It is trivial to see that Q_ ¼ Q^ ¼ QT ¼ QL ¼ Q. Now, consider the class C PS of all power stable copulas (i.e., EV-copulas), the class QPS of all power stable quasi-copulas, and the class APS of all power stable binary aggregation functions. From Proposition 1 we get C PS ¼ C ^PS . This is true due to the closedness of the class P of all Pickands’ dependence functions with respect to suprema. On the other hand, due to Theorem 2, each dependence function generating a power stable quasi-copula Q 2 QPS can be obtained as the infimum of the Pickands dependence functions, i.e., Q can be obtained as the supremum of some copulas. Hence
CPS $ C _PS ¼ QPS ¼ ALPS ; (the last equality is a matter of definition only). As a non-trivial problem, it remains to clarify the position of C TPS , i.e., the class of all aggregation functions coinciding on each track with some power stable copula. Theorem 4. It holds C TPS ¼ C PS . Proof. Obviously, C PS # C TPS . Hence, it is enough to show that each aggregation function A coinciding on each track T with some EV-copula, is also an EV-copula. Choose an arbitrary A 2 C TPS and consider a track T a ¼ fðt; ta Þjt 2 ½0; 1g, for an a 2 0; 1½. Then there is an EV-copula C a 2 C PS such that for all t 2 ½0; 1; Aðt; ta Þ ¼ C a ðt; ta Þ. Thus for each power k 2 0; 1½ it holds a
k
a
k
k
k
Aðt k ; ðt a Þ Þ ¼ Aðt k ; ðt k Þ Þ ¼ C a ðtk ; ðtk Þ Þ ¼ C a ðtk ; ðt a Þ Þ ¼ ðC a ðt; ta ÞÞ ¼ ðAðt; t a ÞÞ : For each ðx; yÞ 2 0; 1½2 there is an a 2 0; 1½ such that y ¼ xa , and hence
Aðxk ; yk Þ ¼ ðAðx; yÞÞk ; i.e., A 2 APS . Moreover, because of the continuity of copulas, A is also continuous. So, we can apply Proposition 2, namely the representation of A in the form (4) with a dependence function d 2 D. Now, consider the track T generated by functions F; G : ½0; 1 ! ½0; 1 given by
FðtÞ ¼ minf1; maxf0; 3t 1gg and GðtÞ ¼ t:
Then there is an EV-copula C generated by a Pickands’ dependence function d 2 P such that AðFðtÞ; GðtÞÞ ¼ CðFðtÞ; GðtÞÞ, i.e., d
ðFðtÞGðtÞÞ for all t 2
1
;2 3 3
log FðtÞ log FðtÞGðtÞ
d
¼ ðFðtÞGðtÞÞ
log FðtÞ log FðtÞGðtÞ
;
. Consequently, for all t 2 13 ; 23 it holds
logð3t 1Þ logð3t 1Þ ¼d : d log tð3t 1Þ log tð3t 1Þ
logð3t1Þ The function f : 13 ; 23 ! R given by f ðtÞ ¼ log is continuous and limt!1þ f ðtÞ ¼ 1, limt!2 f ðtÞ ¼ 0, i.e., for any u 2 0; 1½ tð3t1Þ 3 1 2 3 there is a t 2 3 ; 3 such that u ¼ f ðtÞ and thus for each u 2 0; 1½; dðuÞ ¼ d ðuÞ. The continuity of both d and d results in d ¼ d on ½0; 1, hence, A ¼ C, i.e., A is an EV-copula. h 5. Concluding remarks We have clarified the structure of the class of all binary 1-Lipschitz power stable aggregation functions and the prominent role of power stable quasi-copulas. As the main result we have shown the relationship of power stable quasi-copulas and dependence functions convex at points 0 and 1, satisfying the property dð0Þ ¼ dð1Þ ¼ 1. We have also clarified the
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relationship of EV-copulas and power stable quasi-copulas. As a by-product of the result given in Theorem 4 is the fact that any EV-copula is determined by any of its non-trivial horizontal or vertical sections ha ; v a , respectively, where ha ðxÞ ¼ Cðx; aÞ and v a ðxÞ ¼ Cða; xÞ; a 2 0; 1½. This is not the case of a diagonal section. Indeed, a function d : ½0; 1 ! ½0; 1 is a diagonal section of some EV-copula C if and only if dðxÞ ¼ xa ; a 2 ½1; 2. Up to the trivial cases a ¼ 1 (when C ¼ M) and a ¼ 2 (when C ¼ P), for each a 2 1; 2½ there are infinitely many EV-copulas C such that Cðx; xÞ ¼ xa . Really, any Pickands’
dependence function d 2 P such that d 12 ¼ a2 generates such an EV-copula C. In our further investigation, we open the problem of characterization of 1-Lipschitz power stable aggregation functions of higher dimensions. Acknowledgements The authors kindly acknowledge the support of the grant VEGA 1/0419/13 and the project of Science and Technology Assistance Agency under the contract no. APVV-0073-10. Moreover, the work of R. Mesiar on this paper was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project reg. no. CZ.1.05/1.1.00/ 02.0070. References [1] C. Alsina, R.B. Nelsen, B. Schweizer, On the characterization of a class of binary operations on distributions functions, Stat. Probab. Lett. 17 (1993) 85–89. [2] J. Beirlant, Y. Goegebeur, J. Segers, J. Teugels, Statistics of Extremes: Theory and Applications, John Wiley & Sons Ltd., Chichester, 2004. [3] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer, Heidelberg, Berlin, New York, 2007. [4] F. Durante, R. Mesiar, S. Saminger-Platz, Inform. Sci. 179 (17) (2009). [5] C. Genest, J.J. Quesada Molina, J.A. Rodríguez-Lallena, C. Sempi, A characterization of quasi-copulas, J. Multivariate Anal. 69 (1999) 193–205. [6] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions, Cambridge University Press, Cambridge, 2009. [7] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation functions: means, Inform. Sci. 181 (2011) 1–22. [8] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation functions: construction methods, conjunctive, disjunctive and mixed classes, Inform. Sci. 181 (2011) 23–43. [9] G. Gudendorf, J. Segers, Extreme-value copulas, in: F. Durante et al. (Eds.), Proc. of Workshop on Copulas Theory and its Applications, 2010. [10] A. Kolesárová, R. Mesiar, T. Rückschlossová, Power stable aggregation functions, Fuzzy Sets Syst. 240 (2014) 39–50. [11] R.B. Nelsen, An introduction to copulas, Lecture Notes in Statistics, second ed., vol. 139, Springer, New York, 2006. [12] R.B. Nelsen, M. Úbeda-Flores, The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas, C.R. Acad. Sci. Paris Ser. I 341 (2005) 583–586. [13] G. Salvadori, C. De Michele, N.T. Kottegoda, R. Rosso, Extremes in Nature, An Approach Using Copulas, Springer, Dordrecht, 2007. [14] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231.