on a class of transformations of copulas and quasi-copulas

Fuzzy Sets and Systems 160 (2009) 334–343www.elsevier.com/locate/fss

On a class of transformations of copulas and quasi-copulas

Elisabetta Alvonia,∗, Pier Luigi Papinib, Fabio Spizzichinoc,1

aDepartment of Mathematics for Economic and Social Sciences, University of Bologna, Viale Filopanti, 5, I 40126 Bologna, ItalybDepartment of Mathematics, University of Bologna, I 40126 Bologna, Italy

cDepartment of Mathematics, University “La Sapienza”, I-00185 Rome, Italy

Available online 7 April 2008

Abstract

The theory of copulas is by now a very well established one. Recently, larger classes of functions C : [0, 1]n → [0, 1], that areincreasing in each variable and satisfy some conditions at the boundary (like quasi-copulas), have been the object of fruitful research.Several authors have considered the action of a class of transformations on some aggregation operators, as t-norms, copulas, quasi-copulas and so on. These simple transformations do not preserve in general all properties of copulas (or quasi-copulas): in particular,the fact that only some properties are preserved by these transformations, suggested the introduction of semi-copulas. The purposeof the present contribution is to give a fairly complete picture concerning such action on copulas and quasi-copulas; in particular,we prove results concerning inclusions and strict inclusions among these classes of operators, and those of their transforms.© 2008 Elsevier B.V. All rights reserved.

Keywords: Semi-copulas; Transformations of copulas and quasi-copulas; Construction of semi-copulas; Level curves of survival functions

1. Introduction

Copulas (see e.g. [15] for basic definitions and results) have been studied in details. More recently, larger classes offunctions C : [0, 1]n → [0, 1], that are increasing in each variable and satisfy some conditions at the boundary (likequasi-copulas and semi-copulas) have been studied. The attention has been mainly concentrated on the bivariate casen = 2.

In this context it is very natural, both from the theoretical and the applied point of view, to consider the followingsimple transformations.

Consider continuous, strictly increasing bijections h : [0, 1] → [0, 1] such that h(0) = 0, h(1) = 1; we shall denoteby H the set of all these functions. Denote by B the class of all functions from [0, 1]2 into [0, 1]. Any function h ∈ Hdetermines a transformation, that we will denote by �h : B → B; in fact, given a function S ∈ B, we set

�hS(u, v) = h−1(S(h(u), h(v))). (1)

The transformations we are dealing with have been considered in many papers: see for example [9–11,13]; see also[15, Section 3.3.3]. Such transformations often manifest in different applied fields. Their action on copulas was studiedin [5], and, then, in [2] and in [3] also for semi-copulas.

∗ Corresponding author. Tel.: +39 0512094366.E-mail addresses: [email protected] (E. Alvoni), [email protected] (P.L. Papini), [email protected] (F. Spizzichino).

1 This author was supported by the Italian M.I.U.R. in the frame of the Prin 2004 and Prin 2006 Projects on “Stochastic Methods in Finance”.

0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.03.025

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E. Alvoni et al. / Fuzzy Sets and Systems 160 (2009) 334–343 335

In general, not all properties of copulas (or quasi-copulas) are preserved by these transformations: in particular, thiswas observed in the last two papers; to identify conditions that are preserved by these transformations, the class ofsemi-copulas was introduced.

The action of these transformations is not well behaved and the class of transformed objects has not yet beencharacterized.

Our aim here is to give some new insights on what happens when we transform in this way all copulas or quasi-copulas: we point out that we do not obtain so all semi-copulas. In fact, details about properties the transformed objectspreserve or do not preserve are given.

The plan of this paper goes as follows. In the next section we just recall a few basic definitions and related properties.In Section 3, we recall known results concerning the action of the transformations we are considering. In Section 4 wediscuss the class of objects obtained by transforming quasi-copulas. In Section 5 we describe the operators obtainedby transforming copulas. In Section 6 we give some significant examples. Finally, Section 7 contains a discussion withsome concluding remarks and examples from survival analysis.

2. Definitions and notations

A (bivariate) copula is a function C : [0, 1]2 → [0, 1] satisfying:

C(0, v) = C(u, 0) = 0 for 0�u, v�1, (2)

C(1, v) = v, C(u, 1) = u for 0�u, v�1, (3)

for 0�u�u′ �1, 0�v�v′ �1

C(u′, v′) − C(u, v′)�C(u′, v) − C(u, v). (4)

In particular, condition (4), usually called 2-increasingness, together with (2) implies

C(u, v) is increasing in each variable. (5)

A copula can be seen as the restriction to the unit square of a probability distribution function with uniform marginalson [0, 1]. We shall denote by C the set of all copulas.

A quasi-copula is a function Q : [0, 1]2 → [0, 1] satisfying (3), (5) and

|Q(u′, v′) − Q(u, v)|� |u − u′| + |v − v′|, (6)

for all u, u′, v, v′ in [0, 1].We shall denote by Q the set of quasi-copulas (see [15, Section 6.2], for results on them).Quasi-copulas had been defined originally in a different way; but we preferred to use the above definition, equivalent

to the original one: see [6, Proposition 2].Note that (3), together with (5), implies

C(u, v)� min{u, v} for 0�u, v�1. (5′)

Semi-copulas have been introduced in [1], where ageing functions were considered: these are (symmetric) operators,coming from symmetric elements of C through some �h (h ∈ H); see Section 7 for a discussion concerning ageingfunctions and their relation with the family of the level curves of a bivariate survival function. We just mention thatsemi-copulas had already been considered in a different context, and named t-semi-norms: see [18].

Later semi-copulas were studied, from a technical point of view, in [4].The terminology in those papers is not everywhere consistent; we shall adopt the following terminology.We shall say that a function S : [0, 1]2 → [0, 1] satisfying (3) and (5) is a semi-copula [4]. If moreover S is

continuous, we shall call S a continuous semi-copula.We shall denote by S and SC , respectively, the set of all semi-copulas and the set of all continuous semi-copulas.

336 E. Alvoni et al. / Fuzzy Sets and Systems 160 (2009) 334–343

3. Transformations

As mentioned in the Introduction, we are interested in the following type of transformations. We consider continuous,strictly increasing bijections h : [0, 1] → [0, 1] (so that h(0) = 0; h(1) = 1).

For functions S : [0, 1]2 → [0, 1], h determines a transformation �h defined by (1). We shall denote by �H thisclass of transformations (an element of �H is determined by a function h ∈ H). Given a class A of operators, we shalldenote by �H(A) the class of operators obtained by transforming all elements of A by all elements of �H.

For what follows, it is convenient to recall a few simple, well-known, facts.It is easy to see that properties (2), (3) and (5) are always preserved under the action of a transformation �h ∈ �H.

Therefore, if we start from a quasi-copula Q (in particular, from a copula), then �hQ belongs to SC . Also, it is notdifficult to see that these transformations preserve commutativity and associativity of a map.

Since all functions determining the elements of �H are continuous, then continuous functions: [0, 1]2 → [0, 1] aretransformed into continuous functions. In view of this fact, since we consider transformations of continuous functions,we are interested in the class SC , rather than in S.

Remark 3.1. Composition and inversion of transformations �h just correspond to the same operations on thefunctions h defining them.

Note that S′ = �hS is equivalent to

S = �−1h S′ (with �−1

h determined by h−1 ∈ �H). (1′)

However, it is possible (see the discussion at the end of Section 5) that �h transforms a copula C (or even all copulas)into a copula �hC (respectively: into copulas), while �−1

h does not.We have the following relations (recall that the identity I ∈ �H):

C ⊂ �H(C) ⊂ �H(SC) ⊂ �H(S)

∩ ∩ ‖ ‖Q ⊂ �H(Q) ⊂ SC ⊂ S.

Aim of this paper is to discuss in details some aspects of the above table of inclusions. Note that all the inclusions andsome of the strict inclusions are well known; however, we claim that all the above inclusions hold in the strict sense;in this respect, we prove that the following inclusions are strict:

�H(C) ⊂ �H(Q),

�H(Q) ⊂ SC,

thus also

�H(C) ⊂ �H(SC).

Also, we point out some properties that must be satisfied by the elements in �H(C) or in �H(Q).To conclude this section we note that, starting from two elements �h, �k in �H with h = k, we can easily construct

a copula C such that �hC = �kC: let a be such that

h(a) = b = c = k(a),

it is not a restriction to assume that b < c < 1. Set C(b, b) = b, C(c, c) = c; we have

�kC(a, a) = k−1(C(c, c)) = a = h−1(C(b, b)) = �hC(a, a).

As we shall see in Section 5, there exists also a copula C such that all transformations in �H, when applied to C, givethe same C.


4. Transforming quasi-copulas

It is known that �h ∈ �H and Q ∈ Q do not imply in general �hQ ∈ Q. However, we already noticed that

�H(Q) ⊂ SC = �H(SC).

More precisely, we have (see [3, Theorem 4.1]):

Proposition 4.1. A transformation �h ∈ H is such that �h(Q) ⊆ Q if and only if h is concave.

The problem arises then to find conditions under which a continuous semi-copula is the transformed of a quasi-copula.In this respect we prove the following result.

Theorem 4.1. For a continuous semi-copula S ∈ �H(Q), the following condition holds:

(C1) S(u′, v) = u′ for some u′, v ∈ (0, 1) (u′ �v), implies S(u, v) = u for all u ∈ [0, u′]; a similar fact also holdswith exchanged role of coordinates.

Proof. We prove the first part (the proof of the second part being similar).Take an element S ∈ �H(Q): this means that there exist a quasi-copula Q and a function h ∈ H (thus h and h−1 are

strictly increasing) such that

S(u′, v) = �hQ(u′, v) = h−1(Q(h(u′), h(v)))

or

h(S(u′, v)) = Q(h(u′), h(v)). (7)

Assume that S(u′, v) = u′ for some u′ ∈ (0, 1). Let u ∈ [0, u′]; by using (6) we obtain

h(u′) − h(S(u, v)) = h(S(u′, v)) − h(S(u, v)) = Q(h(u′), h(v)) − Q(h(u), h(v))�h(u′) − h(u)

so

h(S(u, v))�h(u), and then S(u, v)�u,

thus (by (5′)) we have the equality; this proves the theorem. �

Remark 4.1. Theorem 4.1 shows that a necessary condition for a continuous semi-copula S to be in �H(Q) is thefollowing (clearly satisfied by all quasi-copulas):

If S(v, v) = v for some v ∈ (0, 1), then S(u, v) = min{u, v} in [0, v] × {v} and in {v} × [0, v].

The previous theorem, in particular, shows that �H(Q) is strictly contained in SC . In this respect, we give now anexample of a continuous semi-copula violating (C1).

Example 4.1. Construct S(u, v) as follows: we define the level curves in the upper half of [0, 1]2 over the maindiagonal, then we set S(u, v) = S(v, u).

If u� 13 , level curves join [u, 1] to [ 3

2u, 32u].

If 13 �u� 2

3 , level curves join [u, 1] to [ 13 + u/2, 1

3 + u/2].If u� 2

3 , level curves join [u, 1] to [u, u].For this continuous semi-copula, we have: S[ 2

3 , 23 ] = 2

3 ; S[ 49 , 2

3 ] = 13 < 4

9 .Moreover S[ 2

3 , 23 ] − S[ 4

9 , 23 ] = 1

3 > 23 − 4

9 , which shows that S is not 1-Lipschitz.


5. Transforming copulas

We have seen in the previous section that not all elements in �H(Q) are 1-Lipschitz, since in general they are notquasi-copulas. It is known that even �H(C) is not a subset of Q. This fact is also discussed in [3] after Proposition 2.4.For example, consider

W(u, v) = max{0, u + v − 1}and take h(u) = u2. Then, as it is easy to see, W(u, v) is transformed into a function which is not 2-increasing, nor1-Lipschitz; in order to see that these two properties do not hold, check for example, respectively, the quadruplet

(1, 1), (1, 35 ), ( 3

5 , 35 ), ( 3

5 , 1)

and the segment joining ( 35 , 3

5 ) to (1, 35 ).

More precisely, we have the following remark (see [3, Proof of Theorem 3.1]).

Remark 5.1. �h transforms W(u, v) into a quasi-copula if and only if h is concave.So concavity of h is necessary if we want that the transformed objects are in C. More precisely, the following result

is true (see [10,13]).

Proposition 5.1. A transformation �h ∈ �H is such that �h(C) ⊆ C if and only if h is concave.

For conditions on h such that �hC is a copula for a specific copula C, see [5, Theorem 2].The characterization of the maps h such that the copula C(u, v) = uv is transformed into copulas by �h is recalled

in [2, p. 318]: among such maps, we find those determined by the functions h(u) = un, n ∈ N (which are not concavefor n > 1).

The copula

M(u, v) = min{u, v}is the unique copula preserved by all elements in �H (see [11, p. 427]; see also [2, Remark 2.3]): so, in this case,�hM = �h′M also when h = h′.

But something more can be proved. More precisely, we have

Theorem 5.1. Given a copula C(u, v) = M(u, v), there always exists a transformation �h ∈ �H such that �hC isnot a quasi-copula (so it is not a copula).

Proof. First we notice that, if C(u, v) = M(u, v), then there exists u ∈ (0, 1) such that C(u, u) < u. For the proofwe then separately consider two different cases.

Case 1: There exists u ∈ [ 12 , 1) such that C(u, u) < u. Take � > 0 such that C(u+ �, u+ �) = u; define the function

h ∈ H such that h(u + �) = u + �; h(�) = u. We have: Ck(1, u + �) = u + �:

�hC(u + �, u + �) = h−1(C(h(u + �), h(u + �))) = h−1(u) = �.

Thus �hC is neither a copula nor a quasi-copula: in fact it is not 1-Lipschitz on the segment v = u + �.Case 2: Let C(u, u) = u for u� 1

2 and C(u, u) < u for some u < 12 . Let � > 0 so that C(u + �, u + �) = u (clearly

u + � < 12 , so 2u + � < 1).

Define a transformation �h, by using a function h ∈ H such that:

h

(u + 2

3�

)= u + 2

3�, h(u + �) = u + �, h

( �

3

)= u, h(2(u + �)) = �,

where � satisfies C(�, u + �) = u + 23 � (u + � < � < 1).


We have

�hC(u + �, u + �) = h−1(C(u + �, u + �)) = h−1(u) = �

3,

�hC(2u + �, u + �) = h−1(C(�, u + �)) = h−1(u + 23 �) = u + 2

3 �.

Thus �hC is not 1-Lipschitz on the segment y = u + �; so it is neither a copula nor a quasi-copula.We have seen that by transforming quasi-copulas (in particular, copulas), we do not obtain all the elements in SC .

Now we wonder if by transforming copulas, we obtain at least all quasi-copulas.The answer to this question is negative, either. This fact is contained in the following statement.

Theorem 5.2. Assume that S is a continuous semi-copula in �H(C).Then the following must be true:

(C2) if S(u, v′) = S(u′, v′) for some v′ ∈ (0, 1) and a pair u, u′ with u = u′, then S(u, v) = S(u′, v) for everyv ∈ [0, v′]. Also, S(u′, v) = S(u′, v′) implies S(u, v) = S(u, v′) for all u ∈ [0, u′].

Proof. We prove the first part, the proof of the second one being similar.Let S ∈ �H(C). Assume that there exists �h ∈ �H such that

S(u, v) = h−1(C(h(u), h(v)))

for some copula C, or

h(S(u, v)) = C(h(u), h(v)).

Let u�u′ and v�v′. Set

S(u, v) = �, S(u′, v) = �, S(u, v′) = �′, S(u′, v′) = �′.

Then

h(�) = C(h(u), h(v)) = a, h(�) = C(h(u′), h(v)) = b,

h(�′) = C(h(u), h(v′)) = a′, h(�′) = C(h(u′), h(v′)) = b′,

where (C being a copula) b′ − a′ �b − a; therefore

h(�′) − h(�′)�h(�) − h(�).

Thus, if �′ = �′, then � = � (whenever v�v′), which is the thesis. �

Remark 5.2. A simple probabilistic interpretation of Theorem 5.2 will be given in Section 7.

The result we have proved shows that, in particular, if a quasi-copula S satisfies, for a pair u, v with u < v:

(C3) S(u, u) = a < b = S(u, v) = S(v, u) = S(v, v)

then S /∈ �H(C) (since (C2) is not satisfied).For an example of a quasi-copula satisfying (C3), see Example 3.1 in [16]. Another example will be given in the

next section (Example 6.1).We show that condition (C1) (concerning quasi-copulas) is actually weaker than (C2) (see also Example 6.3).

Proposition 5.2. Condition (C2) implies condition (C1).

Proof. We prove this by contradiction. Assume that S does not satisfy (C1): let there exist for example a pair (u′, v) ∈(0, 1) (u′ �v) such that S(u′, v) = u′ and S(u, v) < u for some u ∈ (0, u′). Then we simultaneously have

S(u′, v) = S(u′, 1), S(u, v) < S(u, 1).

so (C2) cannot hold. �


Our results on transformations can also be seen in the opposite way: they indicate when, transforming for examplecontinuous semi-copulas, we may obtain objects with more properties.

Taking into account Remark 3.1, we can point out the following:

Remark 5.3. Assume that h is such that the transformation �−1h determined by h−1 preserves a property; then �hS

can have such property only if S has it.

According to Remark 5.3 and Proposition 5.1, if we start from a continuous semi-copula S which is not a copula,and we take h convex (so that h−1 is concave), then �hS cannot be a copula.

If S ∈ SC\C and h is not convex, it may happen that �hS ∈ C for some h ∈ H.Similar remarks apply to quasi-copulas, according to Proposition 4.1.

6. Some examples

In this section we provide three different examples which clarify the different situations we have described.Our first example is connected with Theorem 5.2.

Example 6.1. A (symmetric) quasi-copula satisfying (C3) can be constructed by defining its level curves correspondingto the value a, for every a ∈ [0, 1], as the union of the four segments:

[a, v], 1 + a

2�v�1;

[u,

1 + a

2

], a�u� 1 + a

2;

[1 + a

2, v

], a�v� 1 + a

2; [u, a], 1 + a

2�x�1.

Moreover, all points in [0, 12 ] × [0, 1

2 ] are set at level 0.In other terms, this is the symmetric copula C, C(u, v) = C(v, u), defined, when v�u, in this way:

C(u, v) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if v�1/2,

u if1 + u

2�v�1,

2v − 1 if u�v� 1 + u

2.

By Theorem 5.2 we already know that �H(Q) (which contains Q) strictly contains �H(C). The next example showssomething more.

Example 6.2. We want to construct a (symmetric) continuous semi-copula S with the following properties:

(a) S belongs to �H(Q);(b) S does not belong to �H(C);(c) S is not a quasi-copula.

Let Q be the quasi-copula of Example 6.1 (satisfying (C3), so Q /∈ �H(C)); then set S = �hQ, where �h is determinedby the function h(u) = u2.

Clearly S ∈ �H(Q).Also, S satisfies (b): in fact, suppose �hQ ∈ �H(C), so S = �kC for some copula C and some k ∈ H. This would

imply (see (1′)): �hQ = �kC, or Q = �−1h (�kC), against Q /∈ �H(C).

Now we prove (c). Consider the values of S = �hQ. We have (recall that h−1(u) = √u):

S( 23 , 2

3 ) =√

Q( 49 , 4

9 ) = 0, S

(√1318 , 2

3

)=

√Q( 13

18 , 49 ) =

√Q(1, 4

9 ) =√

49 = 2

3 .


Therefore S(

√1318 , 2

3 ) − S( 23 , 2

3 ) = 23 >

√1318 − 2

3 .This shows that S /∈ Q, so (c) is true. This concludes the proof. �

We know that by transforming a copula we do not always obtain a copula; also, it is not difficult to single out acontinuous semi-copula satisfying (C1) but not (C2). Next example shows that for some C ∈ C and some �h ∈ �Hwe may obtain a quasi-copula which is not a copula.

Example 6.3. Consider a (symmetric) copula C with the following properties:

C( 13 , 1

3 ) = 0, 12, C( 13 , 2

3 ) = C( 23 , 1

3 ) = 13 , C( 2

3 , 23 ) = 0, 55.

In fact, there exists a copula with this property according to a general result on discrete copulas (see [12]).Take as h ∈ H a piecewise linear function h(u) such that:

h(u) = u for 0�u� 13 and 2

3 �u�1,

h( 54100 ) = 55

100 .

The function h−1 (piecewise linear) is again the identity for 0�u� 13 and for 2

3 �u�1, while

h−1( 55100 ) = 54

100 .

We have

�hC( 13 , 1

3 ) = h−1(C(h( 13 ), h( 1

3 ))) = h−1(C( 13 , 1

3 )) = h−1(0, 12) = 0, 12,

�hC( 13 , 2

3 ) = �hC( 23 , 1

3 ) = h−1(C(h( 13 ), h( 2

3 ))) = h−1( 13 ) = 1

3 ,

�hC( 23 , 2

3 ) = h−1(C(h( 23 ), h( 2

3 ))) = h−1(0, 55) = 0, 54.

From this condition it is easily seen that �hC violates condition (4).On the other hand, it is clear that we may construct a copula taking the values we have indicated for our quadruplet

and such that its transformed by �h is a quasi-copula; therefore �hC ∈ [�H(C) ∩ Q]\C.

7. Discussion and concluding remarks

So far we have pointed out several facts concerning what happens when transformations of the form (1) are applied tocopulas or quasi-copulas. Such transformations are important in many contexts and, in fact, they emerged several timesin the recent literature, for different reasons; we already cited some references. In particular this type of transformationscan be seen as a general tool to generate new t-norms starting from given ones (see [8, Chapter 3]).

We want briefly sketch here also a few different reasons of interest e.g. in the treatment of pairs of continuous randomvariables X, Y.

We consider the case when X, Y are exchangeable and non-negative but our arguments could be extended to moregeneral cases. We, respectively, denote by F and G the joint survival function and the marginal survival function:

F(x, y) = P {X > x, Y > y}, G(x) = F(x, 0) = P {X > x} = P {Y > x}.By Sklar’s theorem we can write

F(x, y) = C[G(x), G(y)],where C, the so-called “survival copula”, is symmetric. For sake of simplicity, we assume G to be strictly decreasing,strictly positive all over [0, ∞) with G(0) = 1. We mentioned already that G is continuous.

1) In different applications (see e.g. [14,17]; see also [3]) transformed models of the following form have beenconsidered:

R(x, y) = �[F(x, y)],where � ∈ H.


If � is such that R is actually a true joint bivariate survival function then the corresponding marginal is obviouslygiven by GR(x) := R(x, 0) = �[G(x)].

A simple computation shows that

CR(u, v) = �(C[�−1(u), �−1(v)]).The conditions that guarantee that CR belongs to C also imply that R is a joint survival function.

2) Let (X1, Y1) and (X2, Y2) be two independent, identically distributed pairs with joint survival function F and set

X = X1 ∧ X2, Y = Y1 ∧ Y2.

Then

FX,Y (x, y) = P {X1 > x, Y1 > y, X2 > x, Y2 > y} = [F(x, y)]2

and the corresponding marginal is

GX(x) = FX,Y (x, 0) = [G(x)]2.

This shows then the interest of considering in particular the transformed model R(x, y) = �[F(x, y)] with �(u) = u2.See also [7] and the references therein.

3) Under our assumptions on G, we can represent F in the form

F(x, y) = G[− log B(e−x, e−y)],where B is the semi-copula defined by

B(u, v) = exp(−G−1[F(− log u, − log v)]).

B has the property to be a semi-copula apt to describe the family of the level curves of F . Actually it is B ∈ �H(C);in fact, as it is easy to check (see [2]) one can write

B(u, v) = h−1[C(h(u), h(v))]with h(u) = G(− log u).

We now consider the condition (C2) for the semi-copula B, i.e. B(u, v′) = B(u′, v′) for some v′ ∈ (0, 1) and someu = u′.

Written in terms of F , the above condition reads F(x, y′) = F(x′, y′) for some y′ > 0 and some x = x′ > 0 (lete.g. be x < x′).

In its turn, the latter identity can also be written in the form P {x < X < x′, Y > y′} = 0 and obviously it impliesF(x, y) = F(x′, y), for all y > y′, that is equivalent to

B(u, v) = B(u′, v) for all v�v′.

This provides then a probabilistic–geometric interpretation of the condition (C2) that must hold for each semi-copulabelonging to �H(C) (see also [14]). On the other hand, belonging to �H(C), B obviously belongs also to �H(Q) andthen it satisfies condition (C1). In terms of F , a direct probabilistic–geometric interpretation can easily be given alsofor the condition (C1).

References

[1] B. Bassan, F. Spizzichino, On some properties of dependence and aging for residual lifetimes in the exchangeable case, in: B.H. Lindqvist,K.A. Doksum (Eds.), Mathematical and Statistical Methods in Reliability, Trondheim, 2002, Series on Quality, Reliability and EngineeringStatistics, Vol. 7, World Science Publication, River Edge, NJ, 2003, pp. 235–249.

[2] B. Bassan, F. Spizzichino, Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes, J. Multivariate Anal.93 (2005) 313–339.

[3] F. Durante, C. Sempi, Copula and semicopula transforms, Internat. J. Math. Math. Sci. 2005 (2005) 645–655.[4] F. Durante, C. Sempi, Semicopulae, Kybernetika 41 (2005) 315–328.


[5] V. Durrleman, A. Nikeghbali, T. Roncalli, A simple transformation of copulas, Technical Report, Groupe Rech. Opér. Crédit Lyonnais, 2000,pp. 1–15.

[6] C. Genest, J.J. Quesada Molina, J.A. Rodriguez Lallena, C. Sempi, A characterization of quasi-copulas, J. Multivariate Anal. 69 (1999)193–205.

[7] C. Genest, L.-P. Rivest, On the multivariate probability integral transformation, Statist. Probab. Lett. 53 (2001) 391–399.[8] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publisher, Dordrecht, 2000.[9] E.P. Klement, R. Mesiar, E. Pap, Transformations of copulas and quasi-copulas, in: M. López Díaz, M.Á. Gil, P. Grzegorzewski, O. Hryniewicz,

J. Lawry (Eds.), Soft Methodology and Random Information Systems, Springer, Berlin, Heidelberg, 2004, pp. 181–188.[10] E.P. Klement, R. Mesiar, E. Pap, Archimax copulas and invariance under transformations, C.R. Math. Acad. Sci. Paris 340 (2005) 755–758.[11] E.P. Klement, R. Mesiar, E. Pap, Transformations of copulas, Kybernetika 41 (2005) 425–434.[12] A. Kolesarova, R. Mesiar, J. Mordelová, C. Sempi, Discrete copulas, IEEE Trans. Fuzzy Systems 14 (2006) 698–705.[13] P.M. Morillas, A method to obtain new copulas from a given one, Metrika 61 (2005) 169–184.[14] G. Nappo, F. Spizzichino, Relations between Kendall distributions and families of VaR curves in bivariate survival models, Technical Report,

Department of Mathematics, University La Sapienza, 2006.[15] R.B. Nelsen, An Introduction to Copulas, second ed., Springer Series in Statistics, Springer, New York, 2006.[16] R.B. Nelsen, J.J. Quesada Molina, J.A. Rodriguez Lallena, M. Úbeda Flores, Some new properties of quasi-copulas, in: L. Rüschendorf, B.

Schweizer, M.D. Taylor (Eds.), Distributions with Fixed Marginals and Statistical Modelling, IMS Lecture Notes in Monograph Series, Vol.28, Institute of Mathematical Statistics, Hayward, CA, 1996, pp. 233–243.

[17] F. Spizzichino, A concept of duality for multivariate exchangeable models, Fuzzy Sets and Systems, Special Issue for the 28th Linz Seminaron Fuzzy Set Theory, this issue, doi: 10.1016/j.fss.2007.10.009.

[18] F. Suárez García, P. Gil Álvarez, Two families of fuzzy integrals, Fuzzy Sets and Systems 18 (1986) 67–81.

http://dx.doi.org/10.1016/j.fss.2007.10.009

on a class of transformations of copulas and quasi-copulas

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