Durante, Fabrizio

English

Università del Salento: ESE - Salento University Publishing

Chapter 6A generalization ofArchimedean copulasIn this chapter, we introduce and study a class of bivariate copulas that depend ontwo univariate functions. This new family, which contains the Archimedean copulas(see section 1.6.4), is presented in section 6.1. Several examples are then provided insection 6.2. Section 6.3 is devoted to the study of the concordance order in our class.The same circle of ideas will also enable us to construct and characterize a new familyof quasi–copulas (section 6.4).The contents of this chapter can be also found in the papers [41, 42].6.1 The new familyWe denote by Φ the class of all functions ϕ : [0, 1]→ [0,+∞] that are continuousand strictly decreasing, and by Φ0 the subset of Φ formed by the functions ϕ thatsatisfy ϕ(1) = 0. Moreover, we denote by Ψ the class of all functions ψ : [0, 1] →[0,+∞] that are continuous, decreasing and such that ψ(1) = 0. Notice that Φ0 ⊂ Ψ.For all (ϕ,ψ) ∈ Φ×Ψ, we introduce the function Cϕ,ψ : [0, 1]2 → [0, 1] defined byCϕ,ψ(x, y) := ϕ[−1] (ϕ(x ∧ y) + ψ(x ∨ y)) . (6.1)Evidently, Cϕ,ψ is symmetric and, by using the properties (1.1) of the pseudo–inverseof a function, it is easily proved that, for all x ∈ [0, 1],Cϕ,ψ(x, 1) = ϕ[−1] (ϕ(x)) = x = Cϕ,ψ(1, x)and0 ≤ Cϕ,ψ(x, 0) = ϕ[−1] (ϕ(0) + ψ(x)) = Cϕ,ψ(0, x) ≤ ϕ[−1] (ϕ(0)) = 0,92 Chap. 6 A generalization of Archimedean copulasviz. Cϕ,ψ satisfies the boundary conditions (C1).Below we shall investigate under which conditions on ϕ and ψ, the function Cϕ,ψdefined by (6.1) is a copula.Theorem 6.1.1. Let ϕ and ψ belong to Φ and to Ψ, respectively, and let C = Cϕ,ψbe the function defined by (6.1). If ϕ is convex and (ψ − ϕ) is increasing in [0, 1],then C is a copula.Proof. Since C satisfies the boundary conditions (C1), it suffices to show that C is2–increasing. Let R = [x1, x2] × [y1, y2] be a rectangle contained in the unit square.We distinguish three cases. If R ⊂ ∆+, thenVC(R) =ϕ[−1] (ϕ(y1) + ψ(x1)) + ϕ[−1] (ϕ(y2) + ψ(x2))− ϕ[−1] (ϕ(y2) + ψ(x1))− ϕ[−1] (ϕ(y1) + ψ(x2)) .Sets1 := ϕ(y1) + ψ(x1), s2 := ϕ(y2) + ψ(x2),t1 := ϕ(y2) + ψ(x1), t2 := ϕ(y1) + ψ(x2).Then (t1, t2) ≺ (s1, s2) and ϕ[−1] is convex, because ϕ is convex. Thus Theorem 1.2.2implies VC(R) ≥ 0.Since C is symmetric, the same argument yields VC(R) ≥ 0 if the rectangle R isentirely contained in the region ∆−.Next, consider the case in which the diagonal of R lies on the diagonal of the unitsquare, viz. x1 = y1 and x2 = y2. If x1 = 0, then VC(R) = ϕ[−1] (ϕ(x2) + ψ(x2)) ≥ 0.Assume then that x1 > 0. We obtainVC(R) =ϕ[−1] (ϕ(x1) + ψ(x1)) + ϕ[−1] (ϕ(x2) + ψ(x2))− ϕ[−1] (ϕ(x1) + ψ(x2))− ϕ[−1] (ϕ(x1) + ψ(x2)) .Now, sets1 := ϕ(x1) + ψ(x1), s2 := ϕ(x2) + ψ(x2), t1 := ϕ(x1) + ψ(x2) =: t2.Since t 7→ (ψ(t)− ϕ(t)) is increasing in [0, 1], we obtainmin{t1, t2} = ϕ(x1) + ψ(x2) ≥ ϕ(x2) + ψ(x2) = min{s1, s2}andt1 + t2 ≥ s1 + s2.Therefore (t1, t2) ≺w (s1, s2), and since ϕ[−1] is convex and decreasing, from Tomic’sTheorem 1.2.3 we have VC(R) ≥ 0. By using the Proposition 1.6.1, we have thedesired assertion.Chap. 6 A generalization of Archimedean copulas 93Remark 6.1.1. Notice that, since t 7→ (ψ(t)− ϕ(t)) is increasing, then ϕ(t) ≥ ψ(t)for all t ∈ [0, 1]. In fact, if there existed x0 ∈ (0, 1) such that ϕ(x0) < ψ(x0), then0 < ψ(x0)− ϕ(x0) ≤ ψ(1)− ϕ(1) = −ϕ(1) ≤ 0,which is a contradiction.If (ϕ,ψ) is a pair of functions that generate a copula C of type (6.1), then, for anyc > 0, also (cϕ, cψ) generates C.Given a copula C of type (6.1) generated by ϕ and ψ, let h and k be the twofunctions given by h(t) := exp(−ϕ(t)) and k(t) := exp(−ψ(t)). Then, we haveC(x, y) = ϕ[−1] (− ln(h(x ∧ y))− ln(k(x ∨ y)))= h[−1] (exp [ln(h(x ∧ y)) + ln(k(x ∨ y))])= h[−1] (h(x ∧ y) · k(x ∨ y)) .In particular, Theorem 6.1.1 can be easily reformulated in a multiplicative form.Theorem 6.1.2. Let h, k be two continuous and increasing functions from [0, 1] into[0, 1] such that k(1) = 1. If h is log–concave and t 7→ h(t)/k(t) is increasing, thenCh,k(x, y) := h[−1] (h(x ∧ y) · k(x ∨ y)) (6.2)is a copula.6.2 ExamplesThe most important family of copulas of type (6.1) is the Archimedean one. Specif-ically, given a function ϕ in Φ0, Cϕ,ϕ is an Archimedean copula with additive generatorϕ. In particular, the copulas Π and W are of this type. On account of this fact, acopula of type (6.1) is called generalized Archimedean copula (briefly GA–copula).Notice that also the copulaM is of type (6.1): it suffices to take ψ = 0 and ϕ ∈ Φ.As a consequence, the family of copulas of type (6.1) is comprehensive, viz. M , Π andW are GA–copulas.Example 6.2.1. Given an increasing and differentiable function f : [0, 1] → [0, 1],let ϕ(t) = − ln t and ψ(t) = − ln f(t) be two functions satisfying the assumptions ofTheorem 6.1.1. The corresponding copula of type (6.1) is given byCϕ,ψ(x, y) := (x ∧ y)f(x ∨ y),which is a member of the family of copulas studied in chapter 4. In fact, notice that,if ψ − ϕ is increasing, then we can deduce that tf ′(t) ≤ f(t) on [0, 1] and, therefore,f satisfies the assumptions of Theorem 4.1.1.94 Chap. 6 A generalization of Archimedean copulasExample 6.2.2. Let δ : [0, 1]→ [0, 1] be in the class D2 of the diagonals of a copula.Take ϕ(t) = 1 − t and ψ(t) = t − δ(t). If ψ is decreasing, then (ψ − ϕ) is increasingand Theorem 6.1.1 ensures that the pair (ϕ,ψ) generates a copula Cϕ,ψ = Cδ givenbyCδ(x, y) := max{0, δ(x ∨ y)− |x− y|} for all x, y ∈ [0, 1].Thus Cδ is a member of the family of MT–copulas, characterized and studied inchapter 5.Example 6.2.3. Take ϕ ∈ Φ0 and ψ(t) = αϕ for α ∈ [0, 1]. Then ψ ∈ Ψ and thecorresponding copula, Cϕ,ψ = Cα, is given byCα(x, y) = ϕ[−1] (ϕ(x ∧ y) + αϕ(x ∨ y))= ϕ[−1]((ϕ(x) + ϕ(y)) ·A(ϕ(x)ϕ(x) + ϕ(y))),whereA(t) =1− (1− α)t, t ∈ [0, 1/2] ;α+ (1− α)t, t ∈ [1/2, 1] .Therefore Cα belongs to the Archimax family of copulas presented in Example 1.6.9.Example 6.2.4. Take ϕ(t) = −αt+α (α ≥ 1) and ψ(t) = 1− t. Then the pair (ϕ,ψ)belongs to Φ×Ψ and satisfies the assumptions of Theorem 6.1.1. The correspondingcopula, Cϕ,ψ = Cα, is given byCα(x, y) = max{0, x ∧ y − 1α(1− x ∨ y)}=α(x ∧ y) + (x ∨ y)− 1α, α(x ∧ y) + (x ∨ y) ≥ 1;0, otherwise.The copula Cα has a probability mass 2α+1 uniformly distributed on the two segmentsconnecting the point(1α+1 ,1α+1)with (0, 1) and (1, 0), respectively, and a probabilitymass α−1α+1 uniformly distributed on the segment joining the point(1α+1 ,1α+1)to (1, 1)(see also page 57 of [114]). In particular, we obtain C1 = W and C∞ = M . Noticethat this class of copulas has been also used in [29].Example 6.2.5. Take ϕ(t) = 1− t and, for α ∈ [0, 1],ψ(t) =α/2, if t ∈ [0, α/2] ;α− t, if t ∈ [α/2, α] ;0, if t ∈ [α, 1] .Chap. 6 A generalization of Archimedean copulas 95Figure 6.1: Support of the copula Cα (α = 2) in Example 6.2.4Then the pair (ϕ,ψ) belongs to Φ×Ψ and satisfies the assumptions of Theorem 6.1.1.The corresponding copula, Cϕ,ψ = Cα, is given byCα(x, y) :=max{0, x+ y − α} if (x, y) ∈ [0, α]2 ;min{x, y} otherwise.Therefore, Cα spreads uniformly the mass on the two segments connecting, respec-tively, the points (1, 1) with (α, α) and (α, 1) with (1, α). Notice that Cα is a memberof the Mayor–Torrens family (1.5).Note that copulas of type (6.1) that are ordinal sums of copulas are characterizedin the followingProposition 6.2.1. The only (non trivial) ordinal sum of copulas that can be ex-pressed in the form (6.1) is the ordinal sum (〈0, a, C〉), where C is a suitable copulaand a ∈ ]0, 1[.Proof. It suffices to observe that the set of idempotent elements of C is given by{0} ∪ [a, 1], where a := inf{t ∈ [0, 1] : ψ(t) = 0}. In fact, given the copula C, let δ beits diagonal section given by δ(t) := C(t, t) = ϕ[−1] (ϕ(t) + ψ(t)). In particular, forall t ∈ ]0, 1[ we have δ(t) < t if, and only if, min{ϕ(t) + ψ(t), ϕ(0)} > ϕ(t), which isequivalent to ψ(t) > 0. Since ψ is decreasing and ψ(1) = 0, we have δ(t) < t if, andonly if, t is in ]0, a[ where a := inf{t ∈ [0, 1] : ψ(t) = 0}.Theorem 6.1.1 highlights the importance of finding generators in order to constructGA–copulas. To this purpose the following result provides useful methods (the proofsare the same given in [61]).Theorem 6.2.1. Let (ϕ,ψ) be a pair in Φ×Ψ. The following statements hold:(a) if f : [0, 1]→ [0, 1] is an increasing and concave bijection, then (ϕ ◦ f, ψ ◦ f) isin Φ×Ψ;96 Chap. 6 A generalization of Archimedean copulas(b) if f : [0,+∞[ → [0,+∞[ is an increasing convex function such that f(0) = 0,then (f ◦ ϕ, f ◦ ψ) is in Φ×Ψ;(c) if 0 < α < 1, then (ϕ(αt)− ϕ(α), ψ(αt)− ψ(α)) is in Φ×Ψ.Notice that additive generators of Archimedean copulas can be combined togetherin order to construct copulas of the type (6.1). In fact, let ϕ and ψ belong to Φ0; inview of Theorem 6.1.1, the convexity of ϕ and the condition that (ψ−ϕ) be increasingensure that Cϕ,ψ is a copula. Consider, for instance, the functions α(t) := 1 − t,β(t) := − ln t and γ(t) := 1/t − 1, which are, respectively, the additive generators ofthe Archimedean copulas W , Π and the Hamacker copulaΠΣ−Π(x, y) :=xyx+ y − xy ;then we obtain the following copulas:Cβ,α(x, y) = (x ∧ y) exp((x ∨ y)− 1),Cγ,α(x, y) =x ∧ y1 + (x ∧ y)− xy ,Cγ,β(x, y) =x ∧ y1− (x ∧ y) ln(x ∨ y) .6.3 Concordance orderThe concordance order between two GA–copulas is determined only by the prop-erties of their generators.Theorem 6.3.1. Let C and D be two GA–copulas generated, respectively, by thepairs (ϕ,ψ) and (γ, η). Let α := ϕ ◦ γ[−1] and β := ψ ◦ η[−1]. Then C ≤ D if, andonly if,α(a+ b) ≤ α(a) + β(b) for all a, b ∈ [0,+∞] . (6.3)Proof. Let x and y be in [0, 1] and suppose, first, that x ≤ y. Then C ≤ D if, andonly if,ϕ[−1] (ϕ(x) + ψ(y)) ≤ γ[−1] (γ(x) + η(y)) .Let γ(x) = a and η(y) = b, then the above inequality is equivalent toϕ[−1](ϕ ◦ γ[−1](a) + ψ ◦ η[−1](b))≤ γ[−1] (a+ b) .Applying the function γ to both sides, we obtainα[−1] (α(a) + β(b)) ≥ a+ b,viz. condition (6.3).If x > y, the proof can be completed by using the same arguments.Chap. 6 A generalization of Archimedean copulas 97Notice that, if C and D are Archimedean copulas generated, respectively, by ϕand γ, then α = β and condition (6.3) is equivalent to the subadditivity of α, asstated in Theorem 4.4.2 of [114].In two particular cases, the concordance order can be expressed in a form simplerthan (6.3).Corollary 6.3.1. Let C and D be two copulas of type (6.1) generated, respectively,by the pairs (ϕ,ψ) and (γ, η). Let α := ϕ ◦ γ[−1] and β := ψ ◦ η[−1].(a) If ϕ = γ is a strictly decreasing function with ϕ(0) = +∞, then C ≤ D if, andonly if, ψ(t) ≥ η(t) for every t ∈ [0, 1].(b) If ψ = η is a strictly decreasing function with ψ(0) = +∞, then C ≤ D if, andonly if, α is 1–Lipschitz.Proof. Since ϕ = γ admits an inverse, α(t) = t. Therefore condition (6.3) is equivalenttob ≤ β(b) = ψ ◦ η[−1](b) for all a, b ∈ [0,+∞] .Taking b := η(t), we have C ≤ D if, and only if, ψ(t) ≥ η(t) for every t ∈ [0, 1].Analogously, for (b), since ψ = η admits an inverse, we have β(t) = t, and (6.3) isequivalent toα(a+ b)− α(a) ≤ b for all a, b ∈ [0,+∞] ,as asserted.6.4 A similar new class of quasi–copulasIt is interesting to ascertain under which conditions the function Cϕ,ψ defined by(6.1) is a quasi–copula; the following theorem provides a characterization of quasi–copulas in the class {Cϕ,ψ : ϕ ∈ Φ, ψ ∈ Ψ}.Theorem 6.4.1. Let ϕ and ψ belong to Φ and to Ψ, respectively. Let Cϕ,ψ be thefunction defined by (6.1). Then Cϕ,ψ is a quasi–copula if, and only if, both the fol-lowing statements hold:(a) for all r ≤ s and t ∈ [0, (ψ ◦ ϕ[−1]) (r)]ϕ[−1](r + t)− ϕ[−1](s+ t) ≤ ϕ[−1](r)− ϕ[−1](s);(b) for all r ≤ s and t ≥ (ϕ ◦ ψ[−1]) (r)ϕ[−1](r + t)− ϕ[−1](s+ t) ≤ ψ[−1](r)− ψ[−1](s).98 Chap. 6 A generalization of Archimedean copulasProof. We already know that, when ϕ and ψ belong to Φ and to Ψ respectively, thefunction C := Cϕ,ψ given by (6.1) satisfies condition (Q1) for a quasi–copula. Thatboth x 7→ C(x, y) and y 7→ C(x, y) are increasing functions for every y ∈ [0, 1] and forevery x ∈ [0, 1], respectively, is an obvious consequence of the fact that both ϕ andψ are decreasing functions. Therefore, in order to complete the proof, it suffices toshow that the assumptions are equivalent to the Lipschitz condition (Q3) for C.Assume, first, that x1 < x2 ≤ y. The inequalityC(x2, y)− C(x1, y) ≤ x2 − x1 (6.4)is equivalent toϕ[−1] (ϕ(x2) + ψ(y))− ϕ[−1] (ϕ(x1) + ψ(y))≤ x2 − x1 = ϕ[−1](ϕ(x2))− ϕ[−1](ϕ(x1)).By setting r := ϕ(x2), s := ϕ(x1) and t := ψ(y), we obtain that t belongs to[0,(ψ ◦ ϕ[−1]) (r)] and r ≤ s; moreover, the last inequality is equivalent to (a).Next assume y ≤ x1 < x2. The inequality (6.4) is equivalent toϕ[−1] (ϕ(y) + ψ(x2))− ϕ[−1] (ϕ(y) + ψ(x1)) ≤ x2 − x1= ψ[−1](ψ(x2))− ψ[−1](ψ(x1)).By setting r := ψ(x2), s := ψ(x1), t := ϕ(y), we have t ≥(ϕ ◦ ψ[−1]) (s) and r ≤ s.For the arbitrariness of s ≥ r, it follows that t ≥ (ϕ ◦ ψ[−1]) (r) and the last inequalityis equivalent to condition (b).The final case, x1 ≤ y ≤ x2, follows from the two previous cases, sinceC(x2, y)− C(x1, y) = C(x2, y)− C(y, y) + C(y, y)− C(x1, y)≤ x2 − y + y − x1 = x2 − x1,which concludes the proof.Although Theorem 6.4.1 characterizes quasi–copulas of the type (6.1), conditions(a) and (b) may be somewhat impractical. However, these conditions are equivalentto the convexity of ϕ, when ϕ = ψ, as is shown in the followingCorollary 6.4.1. Let ϕ belong to Φ0 and let Cϕ,ϕ be a function of the type (6.1).Then Cϕ,ϕ is a quasi–copula if, and only if, ϕ is convex.Proof. By Theorem 6.4.1, Cϕ,ϕ is a quasi–copula if, and only if, for every r ≤ s andfor every t ≥ 0, we haveϕ[−1](r + t)− ϕ[−1](s+ t) ≤ ϕ[−1](r)− ϕ[−1](s),which can be written in the formϕ[−1](r + t) + ϕ[−1](s) ≤ ϕ[−1](s+ t) + ϕ[−1](r). (6.5)Chap. 6 A generalization of Archimedean copulas 99If ϕ is convex, so is ϕ[−1], and therefore (6.5) follows directly from Theorem 1.2.2,observing that (r+ t, s) ≺ (s+ t, r). Conversely, if (6.5) holds, for all a, b ≥ 0 we canputr = a, t =b− a2s =a+ b2.Then, we have2ϕ[−1](a+ b2)≤ ϕ[−1](a) + ϕ[−1](b),viz. ϕ[−1] is mid–convex and, because ϕ[−1] is continuous, it follows that ϕ[−1] isconvex, and hence so is ϕ.The following result provides a sufficient condition for Cϕ,ψ to be a quasi–copula.Proposition 6.4.1. Let ϕ and ψ belong to Φ and to Ψ, respectively. If ϕ is convex,then, for the function Cϕ,ψ defined by (6.1), the following statements are equivalent:(a) Cϕ,ψ is a quasi–copula;(b) for every λ ∈ [ϕ(1), ϕ(0)] the function ρλ :[ϕ[−1](λ), 1]→ R given byρλ(t) := ϕ[−1] (λ+ ψ(t))− tis decreasing.Proof. From Theorem 6.4.1, it suffices to show that C satisfy the 1–Lipschitz condition(Q3). Assume, first, that x1 ≤ x2 ≤ y. The inequalityC(x2, y)− C(x1, y) ≤ x2 − x1 (6.6)is equivalent toϕ[−1] (ϕ(x2) + ψ(y)) + ϕ[−1](ϕ(x1)) ≤ ϕ[−1] (ϕ(x1) + ψ(y)) + ϕ[−1](ϕ(x2)).By setting s1 := ϕ(x2) + ψ(y), s2 := ϕ(x1), t1 := ϕ(x1) + ψ(y) and t2 := ϕ(x2),we have (s1, s2) ≺ (t1, t2) and therefore, since ϕ[−1] is convex, Theorem 1.2.2 ensuresthat (6.6) is satisfied. In this case, the Lipschitz condition is a consequence of theconvexity of ϕ alone.Next assume y ≤ x1 < x2. The inequality (6.6) is equivalent toϕ[−1] (ϕ(y) + ψ(x2))− ϕ[−1] (ϕ(y) + ψ(x1)) ≤ x2 − x1;viz. condition (b).The final case, x1 ≤ y ≤ x2, follows from the two previous cases.Example 6.4.1. Take the functionsϕ(t) := − ln t and ψ(t) := − ln (t+ t2 − t3) .100 Chap. 6 A generalization of Archimedean copulasFor every λ ∈ [0,+∞] the function ρλ : [exp(−λ), 1]→ R is given byρλ(t) := exp(−λ)(t+ t2 − t3)− t.Now, (ϕ,ψ) is in (Φ×Ψ) and ρλ is decreasing, therefore Theorem 6.4.1 ensures thatthe function Cϕ,ψ, given by (6.1) is a quasi–copula. Notice that Cϕ,ψ is not a copula, asshown in Example 4.3.1. This implies that the family {Cϕ,ψ : ϕ ∈ Φ, ψ ∈ Ψ}, where ϕand ψ satisfy conditions (a) and (b) of Theorem 6.4.1, contains proper quasi–copulas.

A generalization of Archimedean copulas

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A generalization of Archimedean copulas

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