Special Issue in memory of Abe Sklar New results on perturbation-based copulas

: A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.


Introduction
The earliest use of what is now called perturbation theory was to deal with otherwise unsolvable mathematical problems of celestial mechanics. When Kepler published his rst law "the orbit of every planet is an ellipse with the Sun at one of the two foci," in [66] and [67, book 5, part 1, III. De Figura Orbitae]) at the beginning of the 17th century, he provided an analytical solution of a classical two-body problem, the two bodies being the planet under consideration and the Sun -in this scenario no perturbation occurred. Many decades later, when three-body problems were studied, e.g., the system Moon-Earth-Sun [59], one observed that the Moon (which has a much smaller mass than both Sun and Earth) does not move along a simple ellipse à la Kepler because of the competing gravitation of the Earth and the Sun, i.e., the constants describing the motion of a planet around the Sun are also in uenced by the motion of other planets and may vary in time. In addition, in the second half of the 19th century the increasing accuracy of astronomical observations also required a higher accuracy of the solutions to Newton's gravitational equations [95] and motivated mathematicians such as Lagrange [80] and Laplace [81] to develop and study fundamental methods of perturbation theory (for a classical survey see, e.g., [9]).
In mathematics, physics, and chemistry, perturbation theory deals with mathematical methods for nding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. Perturbation theory is widely used when the problem at hand does not have a known exact solution, but can be expressed as a "small" change to a known solvable problem.
In quantum mechanics (see [32] and [12]), perturbation theory can be used to describe a complicated unsolved system using a simple, solvable system. Starting with such a simple system whose mathematical solution is known, one may add a weak disturbance to the system (a so-called "Hamiltonian" [7]). If this disturbance is not too large, physical quantities related to the perturbed system (e.g., its energy levels) can be seen as "corrections" to those of the simple system. If these corrections are small compared to the size of the quantities themselves, approximate methods such as asymptotic series can help to calculate them. Now let us turn to the concept of bivariate copulas (introduced in [119], see also [39,64,93]), i.e., to mathematical objects which capture the dependence structure among random variables and which are the topic of our current research. In this context, perturbation usually means that a (small) bivariate function (often called the perturbation factor) is added to a given copula, and one is interested to nd out under which conditions the result is again a copula [36,98]. A prominent example of such a perturbation is the family of Eyraud-Farlie-Gumbel-Morgenstern (or EFGM) copulas given in (2.7), where a parameterized family of perturbation factors is added to the product copula Π (for more details about EFGM copulas and for other families of copulas which can be considered as perturbations see [39,68,93]).
In this paper, we investigate two di erent types of perturbations of general copulas which will be introduced in De nitions 4.1 and 5.8 (where we can identify an interesting class of extremal elements), respectively.
In these constructions we make use of several techniques which are well-known in the theory of copulas, such as x-and y-ippings, the construction of the survival copula, and three variants of the concept of ordinal sums, to name just a few. We study in detail a number of mathematical properties of these perturbations and some interrelations between them. Some of the perturbations discussed here induce interesting extensions of the family of Eyraud-Farlie-Gumbel-Morgenstern copulas (compare [104]). Finally, some relationships to the Markov product of copulas are emphasized, and the values of four dependence parameters (Spearman's rho, Kendall's tau, Blomqvist's beta, and Gini's gamma) of our perturbations are given.

Preliminaries
Copulas are mathematical objects capturing the dependence structure among random variables. The name "copula" for functions linking an n-dimensional distribution and its one-dimensional marginals goes back to Sklar's paper [119] (compare also [120]), where he proved (for the case n = ) a result which is now referred to as Sklar's Theorem. However, links between multivariate distributions and their one-dimensional marginals have been studied before, e.g., by Hoe ding [60,61], Fréchet [50], Dall'Aglio [23][24][25], and Féron [48], and also later on without any reference to the concept of copulas (see, e.g., [108,123] and which is -increasing, i.e., for all (x, y), (x * , y * ) ∈ [ , ] with (x, y) ≤ (x * , y * ) The set of all bivariate copulas will be denoted by C . There are in nitely many elements in C : in the books [63,93] and, more recently, [39] one nds plenty of examples of parametric families (usually with one or two parameters) of copulas, on the one hand, and classes of copulas which can be constructed and characterized by functions in one variable (e.g., by additive and/or multiplicative generators [1,83,111,112] in the case of Archimedean copulas), on the other hand.
In brief, Sklar's Theorem states that, whenever (X, Y) is a random vector with its two marginal distributions F X , F Y : R → [ , ], then there exists a copula C X,Y ∈ C (which is uniquely determined if and only if X and Y are continuous) such that the joint distribution F X,Y : R → [ , ] is given by Conversely, for each copula C ∈ C the function F : R → [ , ] given by is a two-dimensional probability distribution of the random vector (X, Y) such that C X,Y = C. Sklar's Theorem also holds for the general case of n-dimensional probability distributions (proofs for this general case and alternative proofs for the bivariate case can be found in [5, 33-35, 43, 46, 96, 100]).
In the language of Sklar's Theorem the following three basic copulas describe a pair of independent or comonotone dependent or countermonotone dependent random variables X and Y, respectively: the product copula Π :  [30,93]) by, respectively, If X and Y are two continuous random variables and if C X,Y ∈ C is the (unique) copula satisfying (2.1) in Sklar's Theorem then the following stochastic interpretation is an immediate consequence of [93, Theorem 2.4.4]: As an immediate consequence of (2.3), the x-ipping, the y-ipping and the construction of the survival copula (the latter being the composition of x-ipping and y-ipping) are involutive operations on C , i.e., for each copula C ∈ C we have For the three basic copulas W, Π, and M the following relationships can be veri ed easily: A copula C ∈ C which is invariant with respect to the construction of survival copulas, i.e., which satis es C surv = C, is also called radially symmetric [55] (see also [6,8,19]). The three basic copulas W, Π and M are trivial examples of radially symmetric copulas. Given a copula C ∈ C , we sometimes will work with a distinguished section of it, the so-called opposite diagonal section ω C : [ , ] → R de ned by There is an axiomatization for a function ω : [ , ] → [ , ] to be an opposite diagonal section of copulas, and for each such ω there exists at least one C ∈ C such that ω C = ω (see, e.g., [31,45,52,53]). In Section 6, we shall be concerned with several dependence parameters of a copula C ∈ C , in particular with Spearman's rho [121], Kendall's tau [65], Blomqvist's beta [11], and Gini's gamma [56] which can be de ned for each copula C ∈ C and assume their values in the interval [− , ]. The corresponding functions ϱ, τ, β, γ : C → [− , ] are given by (see, e.g., [93]), respectively: C(x, y) dx dy − , τ(C) = [ , ] C(x, y) dC(x, y) − , For the three basic copulas W, Π and M and for each function ξ : C → [− , ] such that ξ ∈ {ϱ, τ, β, γ} we obtain the special values ξ (W) = − , ξ (Π) = , and ξ (M) = .
From Theorems 5.1.1 and 5.1.9 in [93] (see also De nition 5.1.7 in this monograph) it follows that for each copula C ∈ C and for each function ξ : C → [− , ] such that ξ ∈ {ϱ, τ, β, γ} we get for the x-ipping C x ip and the y-ipping C y ip of C and for the survival copula C surv given by (2.2): A particularly interesting and important family of copulas, which is used quite often when the weak dependence of exchangeable random variables should be modeled, is This family was usually referred to as the family of Farlie-Gumbel-Morgenstern copulas [42,58,92,93]. In [39] (see also [13,14,91]) it was pointed out that the corresponding distributions had already been studied in the earlier and, for many years, forgotten work by Eyraud [41]. In recognition of that we will consistently use the name Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas for short) in this paper.
In an analogous way, another type of ordinal sums of copulas based on the lower Fréchet-Hoe ding bound W was suggested in [90] (see also [39]).
A third type of ordinal sum of copulas, the so-called (vertical) Π-ordinal sums, will be used in Section 5. They were originally introduced in [76] as a generalization of some patchwork techniques [28,37,38] and of the gluing of copulas proposed in [118].
In each of the three cases we start with an arbitrary family (]a k , b k [) k∈K of non-empty, pairwise disjoint open subintervals of [ , ] and with an arbitrary family (C k ) k∈K of copulas. Then each of the three functions C Mos , C Wos , C Πos : [ , ] → [ , ] de ned by, respectively, is a well-de ned copula. We call C Mos the M-ordinal sum, C Wos the W-ordinal sum, and C Πos the (vertical) Π-ordinal sum of the summands (]a k , b k [ , C k ) k∈K , and we shall write Another extremal case which is covered by (2.8)-(2.10) is that of an empty index set: the empty ordinal sums M-( a k , b k , C k ) k∈∅ , W-( a k , b k , C k ) k∈∅ and Π-( a k , b k , C k ) k∈∅ coincide with M, W and Π, respectively. Some deeper investigations of these and other types of ordinal sums can be found in [102] (see also, e.g., [38,40,69,70,74,101]).

Some known results on perturbations of copulas
In a number of papers, various perturbations of copulas were introduced and studied from di erent points of view. In most cases, the authors xed a copula D ∈ C and a suitable bivariate function H : [ , ] → R, and looked for conditions on H (and D) guaranteeing that also the function C : [ , ] → R given by was a copula.  was studied in [87] (compare also [77][78][79]86]). (iv) Let f , g : [ , ] → R be suitable functions and put C(x, y) = D(x, y) + f (max(x, y))g(min(x, y)). (3.5) This situation was investigated in [36], while the special case D = Π had already been discussed in [4]. (v) As a special case of (iv), a necessary and su cient condition for the function to be a copula was given in [99, Theorem 2.3], and a simpler su cient condition for being an absolutely continuous copula can be found in Theorem 2.5 in the same paper. Probably the simplest necessary and su cient condition for C being a copula (see [39, Example 1.6.10]) is that the functions f and g are Lipschitz, vanish in and and satisfy f (x)g (y) ≥ − for all (x, y) ∈ [ , ] for which the derivatives exist (compare also [39, Theorem 1.6.9]). In Section 5, we shall present some new results related to this type of perturbations. (vi) As a generalization of (v), the parameterized function was considered for the rst time in the context of copulas in [105] and later in [82,98,117]. A generalized form of (3.7), where in the rst summand on the right-hand side the product was replaced by an arbitrary copula, was discussed in [68]. For a survey and other generalizations of the cases (3.5)-(3.7) see [4].
(vii)For an interesting generalization of (vi), namely,  [72]). The problem, under which conditions on θ and N and N the function is a copula was solved in [47] (see also [86]).
Remark 3.2. Other approaches to perturbations are based on weighted arithmetic and geometric means. For example, the weighted arithmetic mean of two arbitrary copulas is again a copula, and thus we can perturb a given copula D by means of some (arbitrary) copula C putting E = ( − ε)D + εC, where ε ∈ ] , [. In the case of the weighted geometric mean, it is known that the set of extreme-value copulas [57] is closed under this averaging operator. For some deeper study of this kind of problems see [20]. For some other types of perturbation, in particular for those connected with the diagonal expansion of a copula, we refer to [21]. More recently, other perturbations of copulas which are related to random noise were described and investigated in [89,115,116].
The family of EFGM copulas C EFGM θ θ∈[− , ] has many nice properties (see, e.g., [104]), but also some drawbacks. One of them is that the values of the dependence functions of the four dependence parameters given in (2.6) are bounded by − and for each EFGM copula, so only weak dependencies can be modeled by these copulas.
A number of extensions of the family of EFGM copulas has been presented in the literature in order to overcome this constraint. A comprehensive extension (i.e., containing the three basic copulas W, Π and M) was introduced in [62], other approaches were based on quadratic constructions of copulas [75] or on some forms of convexity (such as ultramodularity [85] and Schur concavity [107]), see [103]. A very interesting and natural extension of EFGM copulas are polynomial copulas [122] (in [114] polynomial copulas of degree ve are studied in detail) which can also be seen as special cases of the general perturbation (3.1). Also some of the perturbations investigated in [36,68,77,78,87,98,105] turn out to be extensions of the family of EFGM copulas.

Copula-based perturbations of copulas
We now consider perturbations of an arbitrary copula C ∈ C by means of the opposite diagonal sections given by (2.5) of two copulas C , C ∈ C . The study of these perturbations was motivated by the investigations in [36] or [86,87]. Again, in some special cases we obtain extensions of the family of EFGM copulas.
De nition 4.1. Let C, C and C be three arbitrary copulas and θ ∈ R. Then we consider the function Note that (4.1) is a variant of (3.7) in Remark 3.1(vi): in the rst summand on the right-hand side, the product is replaced by an arbitrary copula C, and in the second summand the functions f and g equal the opposite diagonals (given by (2.5)) ω C and ω C , respectively, of two copulas C and C .
In order to nd out under which conditions the function [C, C , C ] θ is a copula, we have to recall some measure theoretic facts about absolutely continuous and singular parts of copulas (compare [ is absolutely continuous. This function A C is called the absolutely continuous part of the copula C, and the function S C : [ , ] → [ , ] given by S C = C − A C the singular part of the copula C (see [93, (2.4

.1)]). Finally, put
(4. 2) The following result provides a complete solution of the problem under which conditions the function [C, C , C ] θ is a copula.
the absolutely continuous part of C, and α C as given by (4.2). Then the following are equivalent: Proof. Fix an arbitrary copula C and some θ ∈ R, and assume that condition where these derivatives exist. Then each of these functions is the opposite diagonal section of some copula (see, e.g., [31,45]), i.e., there exists a sequence of copulas (Cn) n∈N such that C (x, − x) = f (x) and Cn(x, − x) = gn(x) for each x ∈ [ , ] and all n ∈ N \ { }. For each n ∈ N consider the sets which may be written as and observe that for each n ∈ N λ(An) = λ(Bn) = and If, for some α > α C and for H C,α = {(x, y) ∈ [ , ] | ψ A C (x, y) < α} we have λ H C,α > , then there exist m, n ∈ N such that λ Am ∩ H C,α > and λ Bn ∩ H C,α > . Then, for any (x, y) ∈ Bn ∩ H C,α , the mixed derivative of C as well as the derivatives of f and gn exist such that Clearly, if α − θ < then [C, C , Cn] θ has a negative density on a Borel subset of [ , ] with positive Lebesgue measure and, therefore, cannot be a copula. Thus, necessarily, θ ≤ α. Similarly, [C, C , Cm] θ cannot be a copula whenever α + θ < , and thus θ ≥ −α. Since α > α C was chosen arbitrarily, these two inequalities imply Conversely, x a copula C and some θ ∈ [−α C , α C ], and note that for all copulas C , C ∈ C the two functions f , g : are -Lipschitz and that they satisfy the property f ( ) = f ( ) = g( ) = g( ) = . Also, the inequality ∂x ∂y , f (x) and g (y) exist (this set has Lebesgue measure ) and where the mixed derivative is bounded from below by α C . Moreover, de ne the functions and observe that the following sum of integrals [ , ] ψ A C (x, y) dx dy+ [ , ] f   The following monotonicity properties are an immediate consequence of (4.1): Corollary 4.5. Let C, C , C and D , D be copulas and η, θ ∈ [−α C , α C ]. Then we have: can be considered as an extension of the family C EFGM θ θ∈[− , ] of EFGM copulas, as well as several subfamilies thereof, e.g., As a matter of fact, a copula C is not necessarily symmetric, i.e., we may have C(x, y) ≠ C(y, x) for some (x, y) ∈ [ , ] . In the literature, there are several concepts to measure the asymmetry of a (quasi-)copula (see, for instance, [29] for a recent approach to study the asymmetry of a (quasi-)copula with respect to a curve).
Using the Chebyshev norm, in [71,94] the degree of asymmetry of a copula C ∈ C was de ned as Obviously, a copula C is symmetric if and only if asymm(C) = , and we always have asymm(C) ≤ , the maximal value being attained, e.g., by the copula  In other words, if ω C = ω C and if [C, C , C ] θ is a copula then the construction (4.1) preserves the degree of asymmetry of the copula C. Then ω C (x) · ω C (y) ≠ implies (x, y) ∈ , × , , in which case we have ω C (y) · ω C (x) = . Thus the maximal absolute di erence between the values ω C (x) · ω C (y) and ω C (y) · ω C (x) for (x, y) ∈ [ , ] equals and it is attained at the point , , i.e., for each θ ∈ [− , ] we get for the degree of asymmetry of the copula Π, C , C θ : (4.5) Here the relationship between the parameter θ and the degree of asymmetry of Π, C , C θ is very simple, as it is given by a linear function of |θ|. In Remarks 5.13 and 5.17 we present two families of copulas whose degrees of asymmetry depend on the respective parameters in a more complex way. (iv) If D is an arbitrary copula and if D and D are asymmetric copulas having the same opposite diagonal sections as the two copulas C and C , respectively, which we considered in (iii), i.e., if ω D and ω D coincide with ω C and ω C as given in (4.4) If the copula D is symmetric then in (4.6) the equality holds.

Perturbations of the product copula and ordinal sums
For the copulas considered in this section, recall the constructions of M-ordinal sums and W-ordinal sums in (2.8) and (2.9), respectively, and the x-ipping, the y-ipping and the survival copula of a given ordinal sum de ned in (2.2). The veri cation of the equalities in Lemma 5.1 is a matter of tedious computations.
Lemma 5.1. The relationship between M-ordinal sums M-( a k , b k , C k ) k∈K and W-ordinal sums W-( a k , b k , C k ) k∈K , on the one hand, and the x-ipping, the y-ipping and the survival copula of a given copula, on the other hand, can be formalized as follows: Let us start with a special class of M-ordinal sums constructed by means of two copies of the product copula Π.
If we take into account that Π x ip = Π y ip = Π surv = Π, then the next three equalities follow in a straightforward way from (5.1):  Six copulas Π r ∈ C * Π as given in (5.2) and (5.4)-(5.5) are shown in Figure 1, as well as, in Figure 3, some contour plots of such copulas.
Remark 5.7. If we take into account (2.4), we immediately see that the set C * Π and the corresponding operations (x-ipping, y-ipping and the construction of the survival copula) induce a commutative diagram which is shown in the left-hand part of Figure 2 (compare also Figure 3).
Obviously, this commutative diagram is isomorphic to the commutative diagram in the right-hand part of Figure 2, where we "calculate" only with the parameters of the copulas Π r , due to (5.4)-(5.5). A canonical isomorphism φ : R \ { } → C * Π between these two commutative diagrams is given by φ(r) = Π r . However, the copulas Π r ∈ C * Π have some additional properties. In particular, they are extremal elements (with respect to the usual partial order ≤ for functions from [ , ] to [ , ]) of a special class of copulas which has been studied in a number of papers (see, e.g., [4,36,99,105]).
From Example 1.6.10 in [39] we know that the function C :  Furthermore, de ne the set F Π by g(y)).
Finally, since [39, Example 1.6.10] tells us that each function Π [f ,g] with (f , g) ∈ F Π is a copula, we can consider the partially ordered set C pert,Π of copulas given by With these notations in mind, we are ready to show that each of the copulas Π r with r ∈ ]−∞, [ ∪ ] , ∞[ is an extremal element of the partially ordered set of copulas C pert,Π .
is the greatest function in L b such that the range of its derivative is a subset of [−a, b]. Taking into account that Π [f a,b ,g] ∈ C pert,Π implies f a,b (x)g (y) ≥ − for all (x, y) ∈ [ , ] where the derivatives f a,b (x) and g (y) exist, we see that the function g * /b, /a : [ , ] → R given by is the greatest function in L b such that the range of its derivative is a subset of − b , a . If we put r = b a > , this implies that for each g ∈ L b with (f a,b , g) ∈ F Π we obtain where Π r : [ , ] → [ , ] is given by (5.2). Since for any r , r > with r ≠ r the copulas Π r and Π r are incomparable because of their M-ordinal sum structure, we see that Π r is a maximal element of C pert,Π for each r ∈ ] , ∞[, showing that (i) holds. In order to show (ii), choose an arbitrary r ∈ ] , ∞[. Note rst that if g ∈ L b then for the ipped function g − : [ , ] → [ , ] given by g − (x) = −g( − x) we also have g − ∈ L b and, thus, each Π [f ,g] ∈ C pert,Π , i.e., the set C pert,Π is closed under y-ipping. Since the y-ipping reverses the order and preserves the incomparability of copulas, a copula is maximal if and only if its y-ipping is minimal, showing that for each r ∈ ] , ∞[ the copula Π −r = Π r y ip (compare also (5.2)) is a minimal element of C pert,Π . Proof. Suppose thatû ∈ ] , [ is a non-trivial idempotent element of Π [f ,g] . Then there exist two copulas C and C such that Π [f ,g] is an M-ordinal sum: Therefore, ifû ∈ [x, y] then Π [f ,g] (x, y) = xy + f (x)g(y) = x and, as a consequence, f (x)g(y) = x( − y), i.e., there is a constant b > such that f (x) = bx for all x ∈ ,û and g(x) = −x b for all x ∈ û, . Similarly, ifû ∈ [y, x] then we get f (x)g(y) = ( − x)y, implying that there is a constant a > such that f (x) = a( − x) whenever x ∈ û, , and g(x) = x a whenever x ∈ ,û . Moreover, we have f (û) = bû = a( −û) and g(û) = −û b =û a , both implying a = bû −û . Putting r = −û u > we have a = b r . Summarizing, we have identi ed a solution (fr , gr) ∈ F Π of the functional equation Π [f ,g] (x, y) = xy + f (x)g(y) which is unique up to pairs of positive multiplicative constants (a, a ) and which is given by and Now it is not di cult to check that Π [fr ,gr] = Π r .
In Proposition 5.10 we have seen that, for r ∈ ] , ∞[, the functions fr and gr given by (5.8) generate the copula Π r in a canonical way: Π r = Π [fr ,gr] . Also the copulas Π −r , Π − /r and Π /r can be obtained in a similar way.
Remark 5.12. As a by-product of the second part of the proof of Theorem 5.9, it follows that the set of copulas C pert,Π is closed under y-ipping. However, the set C pert,Π is also closed under x-ipping and the construction of the survival copula: x an arbitrary r ∈ ] , ∞[ and start with the pair of functions (fr , gr) ∈ F Π given by (5.8) = Π /r . Finally, note that we also have the following limit properties: Remark 5.13. Figures 1 and 3 indicate clearly that the set C * Π contains symmetric and asymmetric copulas. More precisely, for each θ ∈ R \ { } the degree of asymmetry asymm(Π r ) of the copula Π r de ned in (4.3) (see [71,94]) is given by (the relationship between the parameter r and asymm(Π r ) is illustrated in Figure 4) For the following family of copulas we shall make use of the so-called (vertical) Π-ordinal sums of copulas given in (2.10) (for more details see [102]).

Markov product and dependence parameters
Several properties of copulas in the classes C * Π given by (5.6) and C pert,Π given by (5.7) can be nicely related to some results on the Markov product of copulas. Finally, we shall discuss the dependence parameters of di erent perturbation-based copulas.

. Perturbations which are idempotent with respect to the Markov product
Given two copulas C , C ∈ C , it is well-known that the partial derivatives ∂C ∂y and ∂C ∂x exist almost everywhere. Therefore the Markov product * : C × C → C (which was introduced as * product in [26], see also [39,73,97] and [93,Section 6.4]) is well-de ned, and the copula C * C : [ , ] → [ , ] is given by It is easy to see that the upper Fréchet-Hoe ding bound M and the product copula Π are the neutral element and the annihilator, respectively, of C with respect to the Markov product, i.e., for all C ∈ C we have Moreover, for each C ∈ C the Markov product of C and the lower Fréchet-Hoe ding bound W are related to the x-and y-ipping and the survival copula of C as follows (see also [93]): Since the Markov product is associative (see [26,Theorem 2.4]), the pair (C , * ) is a monoid, i.e., a semigroup with neutral element: Corollary 6.1. The pair (C pert,Π , * ) is a sub-semigroup, and (C pert,Π ∪ {M}, * ) is a sub-monoid of (C , * ).
Proof. We only have to show that the set C pert,Π is closed under the Markov product * . Choose two arbitrary copulas C , C ∈ C pert,Π such that C = Π [f ,g ] and C = Π [f ,g ] for some pairs of functions (f , g ), (f , g ) ∈ F Π . If we put c (g ,f ) = g (t)f (t) dt then straightforward calculations yield for each (x, y) ∈ [ , ] Note that the constant c (g ,f ) may be split arbitrarily into two positive multiplicative factors, acting on f and g , respectively.

. Dependence parameters
Finally, we shall look for the dependence parameters (Spearman's rho, Kendall's tau, Blomqvist's beta, and Gini's gamma) given in (2.6) of the perturbation-based copulas studied in this paper. First, we will compute the dependence parameters of the perturbations discussed in Section 4 (for the exact formula for [C, C , C ] θ see De nition 4.1). Proposition 6.4. Let C, C and C be copulas and θ ∈ R such that the function [C, C , C ] θ de ned by (4.1) is a copula. For the dependence parameters given in (2.6) we obtain the following formulas: which completes the proof. (iii) If for some C ∈ C the equalities ϱ(C) = τ(C) = hold then we obtain for all copulas C , C ∈ C satisfying ϱ [C, C , C ] θ · τ [C, C , C ] θ ≠ ϱ [C, C , C ] θ : τ [C, C , C ] θ = : .
We can also give some properties of the four dependence parameters related to perturbations discussed in Section 5, in particular to the copulas Π r ∈ C * Π (see (5.2), (5.4) and De nition 5.6) and Π [f ,g] ∈ C pert,Π (see De nition 5.8).
Remark 6.6. For the dependence parameters given in (2.6) we obtain the following formulas (for a visualization see Figure 8): (i) For each r ∈ R \ { } we get ϱ Π r = sign(r) otherwise.
(iii) Consider an open interval Θ ⊆ R with ∈ Θ and a family of copulas (C θ ) θ∈Θ which is continuous with respect to the parameter θ. In [54, Theorem 3.1] (compare also [106]), the authors have shown that, under mild regularity conditions, we have lim θ→ ϱ C θ τ C θ = .
Putting Π = Π we see that the family of copulas (Π r ) r∈R is continuous with respect to the parameter r, and it illustrates the result above because of ϱ Π r : τ Π r = : for each r ≠ . This is the same situation as for the family of Eyraud Proof. Note that, whenever for some pairs of functions (f , g ), (f , g ) ∈ F Π we have then also ξ Π [f ,g ] ≤ ξ Π [f ,g ] for each ξ ∈ {ϱ, τ, β, γ}. Together with Corollary 4.5 and Remark 6.6 this shows that our claim holds.
Still keeping the notations of Proposition 6.7, we can identify the ranges of the dependence parameters for the copulas Π [f ,g] ∈ C pert,Π :

Concluding remarks
Perturbation of functions has a long and multifaceted tradition -also for copulas starting from a copula and perturbing it either by some other function(s) or some other copulas or some derived copulas has been of interest to the scienti c community for di erent reasons. Mostly, the question whether or not or under which conditions the newly built function is again a copula, has been in the focus of investigations. Also in this contribution, we discussed classes of perturbed copulas -starting with some copula C being perturbed by the product of the opposite diagonal sections of two (possibly di erent) copulas C and C . We showed that the search for largest exibility with respect to the choice of the copulas C , C leads to a (possibly rather restricted) interval of possible values for the parameter. However, we also worked with di erent ordinal sums of copulas involving the three basic copulas, showing, e.g., that some particular ordinal sums with the independence copula as its only summand play an important role: they give rise to in nitely many, maximal and minimal elements of a set of perturbation copulas based on the independence copula and two one-place functions of a particular type.
Summarizing we may stress that an emphasis in this contribution has been the identi cation of new structural elements and properties of particular classes of perturbation copulas -revealing new properties, but also revealing new insights by connecting di erent types of perturbation copulas and by connecting our results to results known from di erent areas of copula theory.
For our further research on perturbation of copulas, we think, on the one hand, about the perturbation of copulas of higher dimensions. On the other hand, when considering random variables a ected by some random noise, we see that the original copula, describing the stochastic structure of a random vector, is perturbed into a new copula, and our aim would be an analytical description of the perturbed copula. For some preliminary results in this direction see, e.g., [89].