FABRIZIO DURANTE AND CARLO SEMPI Received 22 April 2004
In memory of Bruno Bassan, friend and colleague
We characterize the transformation, defined for every copula C, by Ch(x,y) := h[−1](C(h(x),h(y))), wherexandybelong to [0, 1] andhis a strictly increasing and con- tinuous function on [0, 1]. We study this transformation also in the class of quasi-copulas and semicopulas.
1. Introduction
The notion of copula was introduced by Sklar [24] who proved the theorem that now bears his name; it is commonly used in probability and statistics (see, for instance, [19, 22,23]). Later, in order to characterize a class of operations on distribution functions that derive from operations on random variables defined on thesameprobability space, Alsina et al. [1] introduced the notion of quasi-copula (see also [12,20,27]). On the con- trary, the notion of semicopula is recent [3,8] and arises from a statistical application:
the study of multivariate aging through the analysis of the Schur concavity of the survival function (see [2,25]). Semicopulas generalize triangular norms (brieflyt-norms), intro- duced by K. Menger in order to extend the triangle inequality from the setting of metric spaces to probabilistic metric spaces, and successfully used in probability theory, mathe- matical statistics, and fuzzy logic [15,22]. We refer to our paper [8] for the properties of semicopulas. Here we recall that a semicopula is a functionS: [0, 1]2→[0, 1] that satisfies the following two conditions:
∀xin [0, 1] S(x, 1)=S(1,x)=x,
S(x,y) is increasing in each place. (1.1) As a consequence of (1.1), given a semicopulaS, one has, for allxandyin [0, 1],
Z(x,y)≤S(x,y)≤M(x,y), (1.2)
whereM(x,y)=min{x,y}and Z(x,y)=
0, (x,y)∈[0, 1[2,
min{x,y}, elsewhere. (1.3)
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:4 (2005) 645–655 DOI:10.1155/IJMMS.2005.645
If a semicopulaCis 2-increasing, namely, for allx,x,y,yin ]0, 1] withx≤x and y≤y,Csatisfies the inequality
C(x,y)−C(x,y)−C(x,y) +C(x,y)≥0, (1.4) then it is a copula (see [19]).
If a semicopulaQsatisfies the 1-Lipschitz condition, namely,
∀x,x,y,y∈[0, 1], Q(x,y)−Q(x,y)≤ |x−x|+|y−y|, (1.5) then it is a quasi-copula.
If a semicopulaTis both commutative
∀x,yin [0, 1], T(x,y)=T(y,x), (1.6) and associative
∀x,y,zin [0, 1], TT(x,y),z=Tx,T(y,z), (1.7) then it is at-norm (see [15,22]).
The classof semicopulas strictly includes the classᏽof quasi-copulas, which, in its turn, strictly includes the classᏯof copulas,Ꮿ⊂ᏽ⊂. Moreover, we will denote by
EandC, respectively, the subsets of commutative (i.e., exchangeable) and continuous semicopulas. The classCstrictly includesᏽandEstrictly includes the set᐀oft-norms (see [8,9]).
Notice that the notion of semicopula is new in a statistical context but is not new in general, since it has appeared in other contexts several times.
The first appearance of which we are aware is in [22, Definition 7.1.5], where the au- thors introduce the setᏵof binary operations on [0, 1] that are nondecreasing in each place and have 1 as the neutral element. By the way, at the same time, they also introduce the subsetᏵCof all the functionsT∈Ᏽthat satisfy (1.5), namely, the set of quasi-copulas!
Then it was again “introduced” in [26] under the name oft-seminorm. Finally, in other words, a semicopula is abinary aggregation operatorwith neutral element 1 [4] or aconjunctor[14].
InSection 2, we will study transformations of semicopulas via a continuous and strictly increasing function on [0, 1]. In Sections3and4, these transformations will be charac- terized, respectively, on the class of copulas and quasi-copulas.
2. The transform of semicopulas
Given a functionh: [0, 1]→[0, 1] that is continuous and strictly increasing withh(1)=1, itspseudoinverseis the functionh[−1]: [0, 1]→[0, 1] defined for allt∈[0, 1] by
h[−1](t) :=
h−1(t), h(0)≤t≤1,
0, 0≤t≤h(0). (2.1)
We denote byΘthe set of all the functionshso defined and we will also consider the sub- setΘiofΘdefined by thoseh∈Θfor whichh(0)=0; the functions inΘiare invertible and the pseudoinverse coincides with the inverse ofh,h[−1]=h−1.
Proposition2.1. For allhandginΘ,
(a)h[−1]is continuous and strictly increasing in[h(0), 1];
(b)for allt∈[0, 1],h[−1](h(t))=tandh(h[−1](t))=max{t,h(0)}; (c) (h◦g)[−1]=g[−1]◦h[−1].
Proof. Statements (a) and (b) are easily proved. In order to prove (c), lethandgbe inΘ.
Then, for allt∈[0, 1], one has (h◦g)[−1](t)=
(h◦g)−1(t), t∈
(h◦g)(0), 1,
0, otherwise,
g[−1]h[−1](t)=
g−1h[−1](t), g(0)≤h[−1](t)≤1,
0, otherwise,
=
g−1h−1(t), t∈D,
0, otherwise,
(2.2)
where
D:= t∈
h(0), 1:g(0)≤h[−1](t)≤1=
(h◦g)(0), 1, (2.3)
which proves assertion (c).
More details on pseudoinverses can be found in [15, Chapter 3]. The following theo- rem is basic for what follows and for the applications.
Theorem2.2. For allh∈ΘandS∈, the functionSh: [0, 1]2→[0, 1], defined, for allx andyin[0, 1], by
Sh(x,y) :=h[−1]Sh(x),h(y), (2.4) is a semicopula. Moreover, ifSis continuous, also its transformShis continuous.
Proof. Iftis in [0, 1], then
Sh(t, 1)=h[−1]Sh(t),h(1)=h[−1]h(t)=t=Sh(1,t). (2.5) Letx,x,ybe in [0, 1] withx≤x. Then
h(x)≤h(x)=⇒Sh(x),h(y)≤Sh(x),h(y)
=⇒h[−1](S(h(x),h(y)))≤h[−1]Sh(x),h(y), (2.6) namely,x→Sh(x,y) is increasing; similarly, one proves that y→Sh(x,y) is increasing.
Theorem 2.2introduces a mappingΨ:×Θ→defined, for allxandyin [0, 1], by Ψ(S,h)(x,y) :=h[−1]Sh(x),h(y). (2.7) We will often set
ΨhS:=Ψ(S,h). (2.8)
The set{Ψh,h∈Θ}is closed with respect to the composition operator◦. Moreover, givenh,g∈Θ, for allS∈, one has
Ψg◦Ψh
S(x,y)=ΨΨ(S,h),g(x,y)=g[−1]ΨhSg(x),g(y)
=g[−1]h[−1]S(h◦g)(x), (h◦g)(y)
=(h◦g)[−1]S(h◦g)(x), (h◦g)(y)=Ψh◦gS(x,y).
(2.9)
The identity mapping in, which coincides withΨid[0,1], is, obviously, the neutral element of the composition operator◦in{Ψh,h∈Θ}. Notice that only ifh∈Θi, doesΨhadmit an inverse function given byΨ−h1=Ψh−1. Notice also that the mappingΨ:×Θi→is theactionof the groupΘion. Moreover, for allh∈Θ, one hasΨhM=MandΨhZ=Z.
Remark 2.3. IfΠ(x,y)=xyis the copula of independence, then, for allh∈Θ,ΨhΠis an Archimedean and continuoust-norm; moreover, the operationΨgives rise to the whole familyᏭCof continuous Archimedeant-norms (written with a multiplicative generator),
ᏭC= ΨhΠ:h∈Θ. (2.10)
We recall that an Archimedeant-normTcan be represented in the form
T(x,y)=g[−1]g(x) +g(y), (2.11) wheregis an additive generator, or in the form
T(x,y)=h[−1]h(x)h(y), (2.12) wherehis a multiplicative generator.
In the class of semicopulas, one can introduce the usual pointwise order: for all S,S∈, one putsS≺SifS(x,y)≤S(x,y), for allx,y∈[0, 1].
Proposition2.4. GivenSandSin, andhinΘ,
(a)the operationΨ is order-preserving in the first place, that is, ifS≺S, thenΨhS≺ ΨhS;
(b)if ΨhS≺ΨhS, thenS(x,y)≤S(x,y)for all(x,y)∈[h(0), 1]2.
Definition 2.5. A subsetᏮofis said to bestable(orclosed) with respect to (or under) Ψif the image ofᏮ×ΘunderΨis contained inᏮ,ΨhᏮ⊂Ꮾfor everyh∈Θ.
It is easily proved that the subsetsEandCare closed underΨ. Moreover, the fol- lowing result can be proved (see also [15,22]).
Proposition2.6. The class᐀of allt-norms is closed underΨ.
Proof. For eachh∈ΘandT∈᐀, it suffices to show that the functionTh:=ΨhT, defined by
∀x,y∈[0, 1] Th(x,y) :=h[−1]Th(x),h(y), (2.13) is associative, namely, it satisfies (1.7). Setδ:=h(0)≥0. Then, ifs,t, anduall belong to [0, 1], simple calculations lead to the following two expressions:
Th
Th(s,t),u=h[−1]TTh(s),h(t)∨δ,h(u), Th
s,Th(t,u)=h[−1]Th(s),Th(t),h(u)∨δ. (2.14) IfT(h(s),h(t))≤δ, then one has
ThTh(s,t),u=h[−1]Tδ,h(u)≤h[−1](δ)=0, (2.15) and either
Th
s,Th(t,u)=h[−1]Th(s),Th(t),h(u)
=h[−1]TTh(s),h(t),h(u)
≤h[−1]Tδ,h(u)≤h[−1](δ)=0
(2.16)
or
Th
s,Th(t,u)=h[−1]Th(s),δ≤h[−1](δ)=0. (2.17) Therefore the associativity equation holds.
IfT[h(s),h(t)]> δ, the considerations are analogous.
The proof of the following proposition is immediate and will therefore not be repro- duced here.
Proposition2.7. The classᏭCis closed underΨ. In particular, ifgis a multiplicative gen- erator of the Archimedean and continuoust-normA, theng◦his a multiplicative generator of ΨhA.
It follows from the definition of the operatorΨthatΨhCis a semicopula for allh∈Θ and for every copulaC∈Ꮿ. However, it is easily checked thatΨhCneed not be a copula.
In order to see this, takeC=Πso thatRemark 2.3ensures thatΨhΠis an Archimedean and continuoust-norm for everyh∈Θ. Now it suffices to recall that at-norm is a copula if, and only if, its additive generator is convex [22, Theorem 6.3.3] and, then, to chooseh in such a way that the corresponding additive generatort→ϕ(t)= −lnh(t) isnotconvex;
thusΨhΠis not a copula. For example, lethbe inΘdefined byh(t) :=t2for allt∈[0, 1].
LetWbe the lower Fr´echet bound defined byW(x,y) :=max{x+y−1, 0}for allx,yin [0, 1]. Then
Wh(x,y)=h−1Wh(x),h(y)=
max x2+y2−1, 0, (2.18)
namely,
Wh(x,y)=
0, x2+y2≤1,
x2+y2−1, otherwise. (2.19)
The functionWhis one of a family oft-norms [22, page 72]. One has Wh
6 10, 1
2
=Wh(1, 1)−Wh
1, 6 10
−Wh 6
10, 1
+Wh 6
10, 6 10
= −2 10<0,
(2.20)
then, in view of [12, Proposition 3],Whis not a quasi-copula.
So, the imageΨhCof a copula should be neither a copula nor a quasi-copula, so that neither the familyᏯof all copulas nor thatᏽof all quasi-copulas are stable underΨ.
3. The transform of copulas
Given a copulaCand a functionh∈Θ, thetransformofCis defined on [0, 1]2by Ch(x,y) :=h[−1]Ch(x),h(y). (3.1) Theorem3.1. For eachh∈Θ, the following statements are equivalent:
(a)his concave;
(b)for every copulaC, the transform (3.1) is a copula.
Proof. (a)⇒(b). It suffices to show thatChsatisfies inequality (1.4). To this end, letx1,y1, x2,y2be points of [0, 1] such thatx1≤x2andy1≤y2. Then the pointssi(i=1, 2, 3, 4), defined by
s1=Chx1
,hy1
, s2=Chx1
,hy2
, s3=Chx2
,hy1
, s4=Chx2
,hy2
, (3.2)
satisfy
s1≤min s2,s3
≤max s2,s3
≤s4, s1−s2−s3+s4≥0. (3.3) By using the notations of [17], one has (s3,s2)≺w(s4,s1), where≺wis theweak majoriza- tion ordering. Becauseh[−1]is convex, continuous, and increasing, it follows from Tomic’s theorem (see [17, (4.B.2)]) that
h[−1]s3
+h[−1]s2
≤h[−1]s4
+h[−1]s1
, (3.4)
namely, inequality (1.4) holds.
(b)⇒(a). It suffices to show thath[−1]is Jensen-convex, that is,
∀s,t∈[0, 1] h[−1]
s+t 2
≤h[−1](s) +h[−1](t)
2 , (3.5)
because, then,h[−1]is convex and, hence,his concave.
Without loss of generality consider the copulaWand pointssandtin [0, 1] withs≤t.
If (s+t)/2 is in [0,h(0)], then (3.5) is immediate. If (s+t)/2 is in ]h(0), 1], then one has W
s+ 1 2 ,s+ 1
2
=s, W t+ 1
2 ,t+ 1 2
=t, W
s+ 1 2 ,t+ 1
2
=s+t
2 =W
t+ 1 2 ,s+ 1
2
.
(3.6)
There are pointsx1andx2in [0, 1] such that hx1
=1 +s
2 , hx2
=1 +t
2 . (3.7)
SinceWhis a copula, it satisfies inequality (1.4):
Whx1,x1
−Whx1,x2
−Whx2,x1
+Whx2,x2
≥0; (3.8)
as a consequence, one has h[−1](s)−h[−1]
s+t 2
−h[−1]
s+t 2
+h[−1](t)≥0, (3.9)
which is the desired conclusion.
The set of concave functions inΘwill be denoted byΘC. It is easy to prove that, for allh,g∈ΘC,λh+ (1−λ)g (λ∈[0, 1]) andh◦g are inΘC. Moreover, ifhis inΘC, then h(tα) and (h(t))αare inΘCfor allα∈]0, 1[. For instance, the following functions are in ΘC:
(a)h(x)=x1/αandh−1(x)=xαwithα≥1;
(b)h(x)=sin(πx/2) andh−1(x)=(2/π) arcsinx;
(c)h(x)=(4/π) arctanxandh−1(x)=tan(πx/4).
Theorem 3.1introduces, for allh∈ΘC, a mapping
Ψh:Ꮿ−→Ꮿ, C−→ΨhC:=Ch, (3.10) which verifies the properties given in the proposition below.
Proposition3.2. The following propertis hold:
(a)for everyhandginΘC,Ψh◦Ψg=Ψg◦h;
(b)if{Cn}is a sequence of copulas that converges pointwise to a copulaCandh∈ΘC, then{Chn}converges pointwise toCh;
(c)Ψh is continuous, in the sense that, for every>0, there existsδ >0such that, for A,B∈Ꮿ,A−B∞< δimpliesΨhA−ΨhB∞<; here
A−B∞:=max A(x,y)−B(x,y): (x,y)∈[0, 1]2; (3.11) (d)Ψhis convex, in the sense that, for every copulasAandBand forλ∈[0, 1],Ψh(λA+
(1−λ)B)≺λΨhA+ (1−λ)ΨhB.
As inSection 2, a subsetᏮofᏯis said to bestablewith respect toΨif the image of Ꮾ×ΘC underΨis contained inᏮ,Ψ(Ꮾ×ΘC)⊂Ꮾ. By using the properties of their generators, it is easily proved that the class of Archimedean and Archimax copulas are stable (for these notions, see [5,11]).
Example 3.3. LetCbe a copula and letrbe a function defined on [0, 1] byr(t)=at+b, witha,b∈]0, 1[,a+b=1. Thenr[−1](t)=max{0, (t−b)/a}and one has
Cr(x,y)=
1 a
Cax+b,ay+b−b, Cax+b,ay+b≥b,
0, otherwise. (3.12)
The copulaCris said to belinear transform ofC.
Remark 3.4. An interesting probabilistic interpretation of formula (3.1) was presented in [13]: if h(t)=t1/n for some n≥1, then Ch is the copula associated with compo- nentwise maxima, X=max(X1,. . .,Xn) andY =max(Y1,. . .,Yn) of a random sample (X1,Y1),. . ., (Xn,Yn) from some arbitrary distribution with underlying copulaC.
Power transformation of copulas was introduced in the theory of extreme value distri- butions [5,6,18]; recently Klement et al. [16] have studied the copulas that are invariant under power transformations and under increasing bijections.
Remark 3.5. LetHbe a bivariate distribution function with unidimensional marginalsF andGand lethbe a strictly increasing function inΘC. From the proof ofTheorem 3.1, it is easily proved that the function
H(x, y)=hH(x,y), (x,y)∈R2, (3.13) is a bivariate distribution function with marginalsh(F) andh(G) and with copulaCh−1. Transformations of type (3.13) were used in the field of insurance pricing [10,28] and they are also calleddistorted probability measuresin the context of nonadditive probabili- ties [7].
We conclude this section with an open problem. LetCbe a fixed copula. What is the subsetΘ(C) ofΘ, depending onC, that ensures thatChis still a copula for allh∈Θ(C)?
For example, ifCis an Archimedean copula with additive generatorϕ, it is easily shown thatChis a copula if, and only if,ϕ◦his convex. In this way, the two following remarks can be useful.
Remark 3.6. For a given copulaC, its transformChmay be a copula even thoughhis not concave. For instance, lethbe the function defined on [0, 1] byh(t)=t2. Thenhis not concave, butΠh=Πis obviously a copula.
Remark 3.7. For a given copulaC, the transformsChandCgmay be equal,Ch=Cg, even though the functionshandgare not equal,h=g. For instance, we consider the copula W and lethbe the function defined on [0, 1] by h(t)=(t+ 1)/2. ThenWh=W and Wid=W, butid=h.
4. The transform of quasi-copulas
Given a quasi-copula (z1,z2)→Q(z1,z2) and a functionh∈Θ, thetransformofQis de- fined on [0, 1]2by
Qhx1,x2
:=h[−1]Qhx1
,hx2
. (4.1)
Lemma4.1. Under the above assumptions,Qhis a quasi-copula if, and only if, for almost all(x1,x2)in[0, 1]2and fori=1, 2,
hxi
·DiQhx1
,hx2
≤hh[−1]Qhx1
,hx2
, (4.2)
whereDiQ=∂Q/∂zi(i=1, 2)exist a.e. on[0, 1].
Proof. For almost all (x1,x2) in [0, 1]2and fori=1, 2, one has DiQhx1,x2
= hxi·DiQhx1
,hx2
hh[−1]Qhx1
,hx2
. (4.3)
SinceQhsatisfies the boundary conditions and is increasing in each place, in view of [21, Theorem 2.1],Qhis a quasi-copula if, and only if,|DiQh| ≤1, namely, if, and only if, the
condition (4.2) holds.
Lemma4.2. Ifhis inΘC, thenQhis a quasi-copula.
Proof. For allx,yin [0, 1], one has
x=h[−1]Qh(x), 1≥h[−1]Qh(x),h(y), (4.4) then, sincehis decreasing a.e. on [0, 1], and since the partial derivatives ofQare smaller than, or equal to, 1,
h(x)·DiQh(x),h(y)≤h(x)≤hh[−1]Qh(x),h(y) (i=1, 2), (4.5)
that is, the condition (4.2).
Connecting the above lemma and the proof ofTheorem 3.1(part (b)⇒(a)), one has the following theorem.
Theorem4.3. For eachh∈Θ, the following statements are equivalent:
(a)his concave;
(b)for every quasi-copulaQ,Qhis a quasi-copula, namely,Ψh:ᏽ→ᏽ.
Acknowledgments
We wish to thank Professor Radko Mesiar who generously showed us the connection of our work with his ongoing research. The topic of the research reported here arose from conversations with B. Bassan and F. Spizzichino; we regret that the untimely death of Bruno Bassan prevents us from thanking him as we should have wished.
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Fabrizio Durante: Dipartimento di Matematica “Ennio De Giorgi,” Universit`a di Lecce, 73100 Lecce, Italy
E-mail address:[email protected]
Carlo Sempi: Dipartimento di Matematica “Ennio De Giorgi,” Universit`a di Lecce, 73100 Lecce, Italy
E-mail address:[email protected]
Special Issue on
Time-Dependent Billiards
Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.
This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://
mts.hindawi.com/according to the following timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
Edson Denis Leonel,Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]
Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;
Hindawi Publishing Corporation http://www.hindawi.com
