Lorenz-Generated Bivariate Archimedean Copulas

An alternative generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a positive random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these "Lorenz copulas", from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An "alchemy" of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.


Introduction
Borrowing tools from the flourishing literature on socio-economic inequality and statistical size distributions [5,12,25], this paper introduces an alternative way to generate a class of non-strict bivariate Archimedean copulas [35], called Lorenz copulas, via the mirrored Lorenz curve [28]. Appealing characteristics of the novel approach are its flexibility and the possibility of characterizing the (upper tail) dependence structure of the related copulas, by studying some simple features of the Lorenz generator, and of its underlying size distribution.
The paper is organized as follows: the next three subsections introduce the basic notation and the necessary tools; Section 2 is devoted to the study of Lorenz-generated non-strict Archimedean copulas; Section 3 contains some examples of copulas generated via the new approach; Section 4 discusses how to obtain new generators and how to develop multiparametric families of copulas; Section 5 closes the paper. The appendix contains some more technical and space-consuming details.

Bivariate Archimedean Copulas: a Quick Review
First introduced by Sklar [46], copulas represent a convenient way of modeling multivariate phenomena [24], by disentangling the joint dependence structure from the marginal behavior. This is particularly true for applications, where the flexibility of copulas appears preferable to the direct fitting of multivariate distributions, which may be difficult to define and deal with [35]. For the sake of completeness, not all statisticians agree with this view: for example Mikosch [33] argues that the static separation of the dependence function from the marginal distributions gives a biased view of stochastic dependence.
In other words, it is possible to represent the bivariate distribution F of the random vector (X1, X2) in terms of the copula function C, and of two uniform margins obtained via the probability integral transform. A bivariate copula is thus nothing more than a bivariate distribution with uniform margins. If the marginals F1 and F2 are continuous then C is unique, otherwise it is uniquely determined on the Cartesian product of the support of the two marginals distributions. It can be easily verified that, for every copula C(u, v), one has where W and M are known as the Fréchet-Hoeffding lower and upper bound respectively [35]. In the bivariate framework here considered, both W and M are proper copula functions. A copula C(u, v) is Archimedean if it is associative, C(u, C(z, w)) = C(C(u, z), w) for every u, z, w ∈ [0, 1], and if its diagonal δC (x) := C(x, x) is such that δC (x) < x for all x ∈ [0, 1].
The associative nature of Archimedean copulas [2] allows for a very convenient representation in terms of a one-place function called the generator.
Note that the pseudo-inverse of a continuous, strictly decreasing function f : [0, 1] → R + with f (1) = 0, is defined as: f [−1] : R + → [0, 1] and such that where f −1 is the standard inverse of f . For a proof, see for example [35]. If one is familiar with non-Newtonian calculus [22], Equation (1) can be recognized as a ϕ-arithmetic operation, a ϕ-sum specifically [31]. This suggests that a bivariate Archimedean copula endows the interval [0, 1] with a semi-group structure.
To an Archimedean copula C(u, v) it is always possible to associate a dual copulâ This dual copula is an S-norm [2], whose generatorφ is an increasing convex function with swapped boundary conditions with respect to those in Theorem 1.
In the literature, several functions ϕ have been proposed over the years, from the well-known logarithmic and exponential generators behind the famous Clayton, Gumbel, Joe and Independence copulas [35], to those based on the inverse Laplace and the Williamson transforms [29]. In particular, this last class of generators provides a solution to the problem of finding d-monotonic functions, thus extending the Archimedean construction to an arbitrary number of dimensions.
Naturally, a large number of generators has given birth to a large number of copulas, and this richness (and flexibility) is one of the reasons of the popularity of the Archimedean family, in particular in applications [14].
From a theoretical point of view, an appealing feature of the Archimedean family is that the properties of a generator essentially determine the properties of the corresponding copula. For example, looking at the value ϕ(0), it is possible to distinguish between strict (ϕ(0) = ∞) and non-strict (ϕ(0) < ∞) copulas [35]. Non-strict copulas are the objects of interest of this paper: their main peculiarity is that a subset of their domain has zero probability mass (but positive Lebesgue measure), taking the name of zero set, or Z(C).
Strictly related to Z(C) is the zero curve v = κ(u), that is the level curve separating the zero set from the part of the copula domain with positive mass. For a non-strict Archimedean copula the zero curve can be easily derived in terms of generator ϕ, by setting C(u, v) = 0 in Equation (1), so that While is not difficult to verify that a non-strict Archimedean copula C(u, v) is not able to model independence directly i.e. C(u, v) = uv = Π(u, v), it can be extremely useful when dealing with phenomena that exhibit upper tail dependence, or when one is interested in the dependence structure of random quantities that do not take on low quantiles at the same time [6,11,26]. In economics, for instance, a situation in which a non-strict copula could be a viable tool for data modeling is given by the presence of minimum production cost (including minimum wages), or the existence of some sort of technological frontier [41].
When dealing with bivariate copulas, and in particular with Archimedean copulas, a very important object of study is the Kendall distribution function K(t) [20]. Such a function represents the bivariate equivalent of the univariate probability integral transform, and it is formally defined as where U and V are standard uniforms on [0, 1]. For a fixed t ∈ [0, 1], the Kendall distribution function can be seen as the measure-also known as the C−measure In an Archimedean copula with differentiable generator ϕ, K(t) can be obtained as with ϕ (t + ) denoting the right derivative of ϕ at t. It is important to recall that for bivariate Archimedean copulas with differentiable generator the function K(t) can be used to determine the corresponding copula via its generator ϕ leading to possible estimation strategies, see [20] for more details. The Kendall distribution function has many applications in copula theory. Just to list two of them that are useful here: 1) K(t) induces a dependence ordering in the set of copulas, the so-called Kendall stochastic ordering [8]; and 2) K(t) can be used to obtain association and dependence measures between random variables. For instance, the well-known Kendall's τ , measuring the concordance between two random variables, can be computed as In this paper τ will represent the main measure of (monotonic) dependence. This is due to the fact that, in Equation (3), τ has a direct link with the generator ϕ via the Kendall distribution function K, something that will be useful in Sections 2 and 3.

The Lorenz Curve
The Lorenz curve L of a non-negative random variable X, with finite expectation and cumulative distribution function F , is defined as where F −1 is the quantile function of X [17]. A Lorenz curve completely characterizes the corresponding distribution function F up to a scale transformation [25]. Introduced by Max Lorenz in 1905 [28], the Lorenz curve is a well-known tool in the study of wealth and income inequality [12]. When the random variable X represents wealth in a given society, the curve L(u) represents the percentage of wealth owned by the lower u% of the population. This makes the curve L essential to study economic size distributions, and to verify the so-called Pareto principle [5,13].
Given a Lorenz curve it is then possible to construct a large number of inequality indices [12]. Among them, a famous one is the Gini G index [21], defined as (5).
Clearly G ∈ [0, 1]. A Gini equal to 0 indicates a society in which everyone possesses the same amount of wealth. A Gini equal to 1 describes the opposite situation: one individual owns everything and all the others nothing. All other values represent intermediate situations: the higher the Gini, the more unequal the society. As observed in [48], all inequality indices are nothing more than generalizations and improvements of some common measures of variability like the variance or the standard deviation. This justifies the rising interest for their application outside inequality studies, in fields like biostatistics and finance [3,15].
The following proposition collects some useful properties of the Lorenz curve which will be needed later. For proofs, please refer to [4]. 3. For all p ∈ [0, 1], the curve L(p) is always bounded from above by LP E (p) = p and from below by LP I (p) = 0 for all p ∈ [0, 1) and LP I (1) = 1. The curves LP E (p) and LP I (p) are respectively called perfect equality and perfect inequality lines.
4. L can be seen as a distribution function. In particular, it represents the distribution function of the random variable YL = L −1 (U ) with U standard uniform on [0, 1].
From Proposition 1 one can derive two important facts: 1) every non-decreasing convex function g : [0, 1] → [0, 1], such that g(0) = 0 and g(1) = 1 is a Lorenz curve [48] corresponding to some non-negative random variable X with finite expectation; 2) every Lorenz curve can be seen and used as a distortion function [7,30], i.e. an integral transform generating increasing convex functions in the unit square. This second fact will prove useful later in the paper.

Orders
Dependence orderings are multivariate stochastic orders defining posets among copulas [42]. The following definition introduces three cases relevant for the paper. Definition 1. Let C1(u, v) and C2(u, v) be two copulas, with Kendall distribution functions K1(t) and K2(t) respectively. Denote by xC (v), the conditional probability P (V ≤ v |U ≤ x), and by xC −1 its inverse. We have If C1(u, v) and C2(u, v) are Archimedean, one has the following relevant implications LT D ⇒ P K ⇒ P QD.
As proven in [6], for Archimedean copulas the conditions given in Definition 1 can be restated in terms of their generators, as in the following theorem. (y)) is super-additive.
As in the case of copulas, it is possible to define notions of stochastic ordering which make the set of nonnegative random variables with finite expectations a poset [4]. The following definition introduces two useful stochastic orders related to the Lorenz curve.
Definition 2. Let X1 and X2 be two non-negative random variables with finite expectation, and let L1, L2 be their Lorenz curves. The Lorenz order and the star order are defined as follows: Observe that, if X1 L X2, then G1 ≥ G2, where G1 and G2 are the Gini indices of X1 and X2 respectively. Furthermore, notice that the star order condition in Definition 2 is not the usual one relying on quantile functions [42]. However, as shown in the appendix, the two definitions are equivalent. Finally, it can be easily shown that the star order implies the Lorenz order [4].

Lorenz Generators and Lorenz Copulas
Let L be a strictly increasing Lorenz function, associated to the strictly increasing distribution function F of a non-negative random variable X with finite mean. For p ∈ [0, 1] define the mirrored Lorenz curve as This new function is strictly decreasing and convex, withL(0) = 1 andL(1) = 0.
Recalling Theorem 1, it is evident thatL(p) is a valid generator for a bivariate Archimedean copula C(u, v). SinceL(0) < ∞, the copula will be non-strict. Similarly, the original Lorenz curve L(p) can be seen as a proper generator for the dual copulaĈ(u, v).
The Gini index corresponding toL is given byḠ = 2 1 0 (1 − p −L(p))dp, and it is easy to verify that its value will coincide with that of the standard Gini G associated to L. For this reason, the notation G will be used to indicate both indices.
Definition 3. Let L andL be the standard and the mirrored Lorenz curves associated to a non-negative random variable X with finite mean. The corresponding bivariate non-strict Archimedean copula C(u, v) is given by C(u, v) is also referred to as the copula associated with X, or the copula generated by L andL.
In the rest of the paper, C(u, v) will represent a non-strict bivariate Lorenz-generated Archimedean copula, often just called Lorenz copula for brevity.
Every non-strict copula is characterized by the presence of a zero set and the relative zero curve. For a Lorenz copula, the zero curve κ(p) is easily derived to be Interestingly, the function κ(p) is itself a mirrored Lorenz curve, given that it follows the composition rules described in [4]. This means that a particular Gini index, the zero Gini Gκ, can be computed from it. Such an index has an appealing interpretation in terms of the corresponding Lorenz copula: Gκ measures indeed how far C(u, v) is from its Fréchet-Hoeffding bounds, i.e. from co-and countermonotonicity. Gκ → 0 indicates that the copula is in the limit its Fréchet-Hoeffding upper bound M , while Gκ = 1 indicates that C(u, v) coincides with the Fréchet-Hoeffding lower bound W .
The following proposition clarifies the relation between Gκ and the Gini G associated with the Lorenz generator L.
Proposition 2. Let Gκ be the zero Gini of the Lorenz copula C(u, v), while G is the Gini associated to the Lorenz generatorL. Then Gκ ≥ G.
Moreover, if X1 and X2 are two non-negative random variables and X1 L X2, one has where G i κ , i = 1, 2, is the zero Gini of the copula associated with Xi.
Proof. To prove the first statement, notice that In order for Gκ ≥ G to hold true it is sufficient to show that the right-hand side of Equation (8) is non-negative.
To prove the second statement it is sufficient to show that where Li(p) is the Lorenz curve of Xi, i = 1, 2, such that L1(p) ≤ L2(p) for every p ∈ [0, 1]. The function 1 − L(1 − p) is known as Leimkuhler curve in the inequality literature [38] and it induces a partial ordering, LK , in the space of non-negative random variables and size distributions [25]. According to such an order, X1 L X2 if and only if X1 ≺ LK X2. Now, observing that L −1 (p) is an increasing concave function for p ∈ [0, 1], and that L −1 1 (p) ≤ L −1 2 (p) for all p ∈ [0, 1] thanks to the Leimkuhler order, then we conclude that (10) holds true.
From Proposition 2 one derives that a necessary condition for G 1 κ ≥ G 2 κ is that G1 ≥ G2. Such a result proves extremely useful when generating Lorenz copulas that need to satisfy some specific conditions in terms of their zero sets and zero curves; more details in Section 3.
Many analytic properties of the copula C(u, v) trace back to the non-negative finite-mean variable X, its Lorenz curve L and the associated mirrored Lorenz generatorL. In the following subsections these properties are collected per topic.

Bounds and singularities
The next proposition clarifies under which conditions a Lorenz copula replicates the Fréchet-Hoeffding lower bound.
Proposition 3. Let C(u, v) be a non-strict Archimedean copula, whileL is its Lorenz generator obtained from the curve L of the non-negative finite-mean random variable X. The following statements are then equivalent: Proof. The goal is to show that 1 ⇔ 2 and 2 ⇔ 3.
Assume that L(p) = LP E (p). The mirrored Lorenz isLP E (p) = 1 − p. By applying Definition 3, one gets To prove that 2 implies 3, it is sufficient to compute the Gini index in Equation (5) for L(p) = p, obtaining G = 0.
Finally 3 ⇒ 2 holds since L(p) ≤ p, therefore the only solution for the functional equation 1 0 p − L(p) dp = 0 is L(p) = p, which concludes the proof.
Regarding the Fréchet-Hoeffding upper bound, no Archimedean structure is able to replicate it exactly [35]. Therefore, an if-and-only-if characterization cannot be given either for a Lorenz copula. However, it is possible to show that, if a Lorenz curve converges point-wise to the perfect inequality case, and thus its Gini tends to 1, then the corresponding C(u, v) will have the upper bound M as its limit.
Proposition 4. Let C(u, v) be the Lorenz copula generated by L and such that C(u, v) → M (u, v) = min(u, v). Then L → LP I (orL →LP I ) and G → 1.
Proof. A necessary and sufficient condition for an Archimedean copula with generator ϕ θ parametrized by θ to attain the Fréchet-Hoeffding upper bound is given in [35], i.e.
In terms of Lorenz generators, for a Lorenz curve L θ of parameter θ, such a condition clearly becomes The only Lorenz satisfying such a condition is the perfect inequality one, LP I , the lower bound for every Lorenz curve, as per Proposition 1. In fact, all the other Lorenz curves are increasing and bounded, hence they cannot satisfy lim θ→θ * L θ (p) = ∞ for every p ∈ (0, 1), which is the only condition for Equation (12) to hold for a generic (non perfect inequality) Lorenz with lim θ→θ * L θ (p) = 0, for p ∈ (0, 1). This said, by Equation (5) the Gini index associated to the Lorenz generator will converge to 1, the value for perfect inequality.
Following Propositions 3 and 4, a Lorenz copula may attain the Fréchet-Hoeffding bounds if the distribution of the underlying variable X can model both a completely even and a totally uneven distribution of wealth. As discussed in [25], this is not always the case, meaning that not all Lorenz copulas can reach the bounds W and M .
Another important feature of non-strict Archimedean copulas is the presence or the absence of a singular part. For a Lorenz copula this completely depends on the continuity of the quantile function of the underlying random variable X.
Proposition 5. Let C(u, v) be the Lorenz copula associated to the non-negative finite-mean random variable X. Then C(u, v) exhibits a singular component if X has a continuous density and a bounded support. Furthermore the singular component is placed on the zero curve.
Proof. Since X has a continuous density, the Lorenz curve L is twice differentiable, and so is the generatorL. Hence, if present, any singular part must be located on the zero curve [35].
In order to quantify the mass of the singular part, one needs the Kendall distribution function of the Lorenz copula C(u, v), which is Thanks to Theorem 4.3.3 in [35], the C-measure of the zero curve is given by where b ≤ +∞ is the upper bound of the support of X and µ = E[X]. If X has no finite upper bound, clearly b = +∞, and the C−measure of the zero curve is zero, thus concluding the proof.
If the random variable X does not have a continuous density, Proposition 5 does not hold. However, one can easily verify that, by taking continuous non-differentiable Lorenz curves, as those arising from discrete random variables, it is possible to define Lorenz copulas with singular parts placed also in other locations of the copula support, not necessarily on the zero curve. The localization and the quantification of the C−measure for these singularities is quite simple, and it depends on the jumps in the quantile function of X. For a jump in t, the C−measure is in fact µ

Dependence and inequality orders
In the theory of copulas, a fundamental topic is the analysis of the dependence structure induced by a given copula function. For Lorenz copulas the dependence structure is directly connected to the underlying variable X and to the stochastic orders discussed in Subsection 1.3. Proposition 6. Let C1 and C2 be two Lorenz copulas associated to X1 and X2, non-negative random variables with finite means. One has that C1 LTD C2 if and only if X1 * X2.
Proof. The proof is a straightforward application of Definition 2 and Theorem 2. In particular, note that the condition on the generators for the LTD order, ≥ 0, can be rewritten in terms of the Lorenz generators as ≥ 0, which is exactly the condition for X1 * X2 according to Definition 2.
If two Lorenz copulas are left tail ordered, then one of the associated random variables is more unequal than the other. In terms of Gini indices, one can easily verify that a necessary condition for C1 LTD C2 is that G1 ≥ G2.
Proposition 7. Let C1 and C2 be two Lorenz copulas associated with the non-negative finite-mean random variables X1 and X2, with Gini indices G1 and G2. Then C1 PK C2 only if G1 ≥ G2.
Proof. By Theorem 2 a necessary and sufficient condition for C1 PK C2 is L1(L [−1] 2 (u)) being star-shaped. Looking at derivatives, the star shape condition implies that L 1 (u) L 2 (u) is increasing. This is equivalent to YL 2 * YL 1 , where YL ∼ L (recall the last point in Proposition 1). Therefore, remembering that the star order implies the Lorenz one [4], and setting L (1) (p) = p 0 L(t)dt, one gets that a necessary condition for the Kendall order is that Equation (15) can be rewritten in terms of the original quantile functions, i.e.
This condition represents an ordering on non-negative random variables called inverse stochastic dominance of third degree (ISD (3)) between X1 and X2, and it can be shown [34] that the condition G1 ≥ G2 is a necessary one for ISD(3) to hold.
Here below a graphical summary of the relations among the orders cited in this section is provided.
As stated, the LTD and * orders imply each other. LTD then implies PK and PQD, and from PQD one can derive that if C1(u, v) P QD C2(u, v), then τ1 > τ2. Similarly, * implies L, which implies ISD(3). From ISD(3) one can finally obtain an order on the Gini indices.

Upper tail dependence
An interesting aspect of Lorenz copulas is the possibility of having different types of tail dependence. Once again, the underlying variable X plays a major role.
In [9], the upper tail behavior of Archimedean copulas is classified into three regimes, for which a characterization is offered in terms of generators:

Tail independence, if and only if
ϕ (1) < 0. ϕ(1−t) > 1. For Lorenz copulas, a necessary and sufficient condition for tail independence is that the support of the underlying non-negative finite-mean variable X has a lower bound strictly larger than zero. In fact, it is sufficient to observe that Equation (17) becomesL where F −1 is the quantile function of X and µ = E[X] > 0. Looking at the right-hand side of Equation (18) the condition on the lower bound is then evident. According to the conditions in [9], asymptotic upper tail dependence requiresL (1) = 0, therefore the lower bound of X needs to be 0: it cannot be negative, since X is non-negative, and it cannot be larger than 0, or it would mean tail independence.
In order to decide whether the upper tail dependence is asymptotic or not, one needs to study the behavior of limp→0 − pL (1−p) L(1−p) . In terms of X and of its cumulative distribution function F , if one sets y = F −1 (x) and applies L'Hôpital's rule twice, such a limit becomes lim y→0 F (y)F (y) (F (y)) 2 ≤ 1.
If the limit above is strictly smaller than 1, one has tail dependence, while the asymptotic case appears when the limit is exactly 1.
Theorem 3. A Lorenz copula shows asymptotic tail dependence when limy→0 F (y)F (y) (F (y)) 2 = 1, that is if and only if the underlying X has support starting in zero together with a lognormal-like left tail in a neighbourhood of zero.
Proof. To prove the if part, it is sufficient to show that the lognormal distribution attains the equality in (19).
To prove the only if part, assume that there exists another Lorenz curve L(x), which is not the one of a lognormal random variable, and such that lim p→0 p L (p) If such a Lorenz curve exists then it must be true that where L(p) is the Lorenz curve of a lognormally distributed random variable with parameter σ > 0, i.e. L(p) = Φ(Φ −1 (p) − σ). Now, recalling the definition of limit we know that it exists an arbitrary small such that log L(p) log L(p) Therefore, in a neighbourhood of zero then the following differential equation should hold: where L (p) = e −σ( 1 2 σ−Φ −1 (p)) . Solving Equation (24), one finds L(p) Φ(Φ −1 (p) − σ), which is the lognormal Lorenz curve up to a constant. Therefore by the uniqueness of the solution of a differential equation (up to a shift), one can conclude that, to attain equality in Equation (19), a lognormal behavior in the vicinity of zero is needed.

Examples of Lorenz Copulas
The present section is devoted to the illustration of some Lorenz copulas. Playing with Lorenz generators, it is not only possible to recover very well-known models, but-more interestingly-one can obtain new non-strict Archimedean copulas, with useful tail properties.

The Lognormal Lorenz Copula
The Lognormal Lorenz copula is obtained by assuming X to be lognormally distributed, so that the generator is where Φ is the distribution function of a standard normal distribution, Φ −1 the corresponding quantile function, and σ > 0 is a scale parameter (equal to the standard deviation of log(X)). Figure 1 shows examples of the lognormal generator for different values of σ.
Notice that the quantity Φ(Φ −1 (y)−σ), with σ > 0, is a well-known distortion function in actuarial mathematics, usually called Wang transform, and it has powerful applications in the fields of asset pricing, risk theory and utility theory [30,47]. Furthermore, in terms of non-Newtonian calculus [22], it represents the pseudo-difference of a variable y and a constant σ: just notice that, given the continuity of Φ −1 , one has σ = Φ −1 (Φ(σ)), so that [31,36].
Using Equation (6), the functional form of the lognormal Lorenz copula is: Figure 2 shows the surface of CLN (u, v) for σ = 0.5 and σ = 2.
Since the lognormal variable X has an unbounded support (its right-end point is xF = +∞), Proposition 5 guarantees that the lognormal Lorenz copula has no singular part. Moreover, Theorem 3 tells us that such a copula is characterized by asymptotic tail dependence, whose strength grows with σ. In Figure 3 two simulations are given, and in both of them it is possible to notice the expected tail behavior: observations in the top right corner are more dependent. The Kendall distribution function associated to CLN (u, v) is given by: which is trivially obtained by noting that the quantile function of a lognormal distribution rescaled by the mean is given by e −( σ 2 2 −Φ −1 (1−t)) . From Equation (27), one can then obtain the value of the Kendall's τ , but this is only possible numerically, the analytical derivation being unfeasible. In Figure 4 the relation between τ and the parameter σ is presented. It is clear that monotonic dependence grows with σ. This is somehow expected by looking back at Figure 3, where not only tail dependence gets stronger as σ becomes larger, but the size of the zero set decreases. In the limit, for σ → ∞, the lognormal Lorenz copula tends to the Fréchet-Hoeffding upper bound M (and to W for σ → 0).
Finally, some considerations in terms of stochastic orders. It is known that the lognormal distribution is star ordered in σ [25]. Namely, if σ1 > σ2 then X1 * X2, with Xi ∼ LN (µ, σi), i = 1, 2. Thanks to Proposition 6 this means that the lognormal Lorenz copula is LTD ordered, and this also implies the positive quadrant dependence and the Kendall order.

The Shifted Exponential Lorenz Copula
The shifted exponential Lorenz copula is obtained via the generator with g ∈ (0, 1 2 ]. When g = 1 2 the mirrored Lorenz curve in Equation (28) corresponds to that of a standard exponential random variable X ∼ Exp(λ). Notice that, for all exponentials, the (mirrored) Lorenz curve does not depend on λ, i.e. all exponentials share the same Lorenz curve, as observed in [15]. For g < 1 2 the random variable X is shifted away from zero by a factor equal to (1 − 2g)λ. Figure 5 shows some examples of the generator LSE(p) for different values of g.   The shifted exponential Lorenz copula obtained from Equation (28) is where W−1 is the lower-branch of the Lambert W function [1]. Figure 6 presents two examples of CSE(u, v), for g = 0.5 and g = 0.2. In the appendix, the details of the derivation of Equation (29) are presented. The copula CSE(u, v) is characterized by some relevant properties and facts, which we can list as follows: Figure 6: Surfaces of a shifted exponential Lorenz copula for different values of g.
1. The shifted exponential Lorenz copula has no singular part. This is a consequence of the unbounded support of the exponential distribution.
2. The shifted Exponential random variables are star ordered with g [23]. Therefore, by Proposition 6, shifted exponential Lorenz copulas are ordered according to the LTD dependence order.
3. Since the Lorenz curve of perfect inequality is never attained by a shifted exponential random variable [25], Proposition 4 suggests that CSE(u, v) is always bounded away from the Fréchet-Hoeffding upper bound M .
4. The Kendall's τSE of the shifted exponential Lorenz copula cannot be written in closed form, but only in terms of Gamma and Exponential Integral functions [1]. However, it can be easily evaluated numerically. Figure 7 shows its behaviour as a function of the parameter g. Interestingly, its range of variation is [-1,0.227], in line with the previous point.
5. The copula exhibits tail dependence only when g = 1 2 , i.e. when the support of the underlying random variable starts in 0. For all the other values of g, CSE(u, v) is tail independent. Figure 8 shows two simulations from CSE(u, v), with g = 0.5 and g = 0.2. As expected, the former case shows tail dependence, while the latter manifests tail independence.

The Pareto Lorenz Copula
The Pareto Lorenz copula emerges when X is Pareto distributed with shape/tail parameter α and scale xm > 0, with mirrored Lorenz curve equal toL For a Pareto random variable, α > 1 is required in order to guarantee that E[X] < ∞, so that the Lorenz curve is defined. In Figure 9 some examples of LP for varying α are given. The Pareto Lorenz copula is It is worth noticing that, by setting θ = 1 α − 1, the Pareto Lorenz copula coincides with the non-strict Clayton family, obtained for θ ∈ (−1, 0) [35].
Since the support of a Pareto random variable starts in xm > 0, the Pareto Lorenz copula is upper tail independent for every choice of α. Moreover, since X ∼ P areto(α, xm) is unbounded from above, CP (u, v) has no singular component.
As shown in the appendix, the Pareto Lorenz copula is LTD ordered, as expected being a subset of the Clayton family. Recall that LTD then implies PK and PQD.
Finally, it is interesting to look at the role of the tail parameter α in the Kendall's τ of CP (u, v). One can easily verify that Equation (32) can be re-written in terms of the Paretian Gini index GP = 1 2α−1 [25], getting Equation (33) shows that τP is an increasing function of GP ∈ [0, 1], moving from -1 towards 0. This implies that the intensity of the association between two random variables coupled with a Pareto Lorenz copula decreases-in absolute value-with an increase in the inequality (in socio-economic terms) of the underlying Pareto random variable X.
Besides the Paretian case, it is worth stressing that, in general, the connection between τ and G is always rather interesting in Lorenz copulas.

The Uniform Lorenz Copula
The Uniform Lorenz Copula represents another interesting case. The underlying non-negative finite-mean variable X is taken to be uniformly distributed on [a, b], with 0 ≤ a < b < ∞. One has The uniform Lorenz copula is and two examples of surfaces are given in Figure 11. As far as the properties of CU (u, v) are concerned, one can observe the following: 1. The uniform Lorenz copula always possesses a singular part. From Equation (14), the C−measure is a+b 2b > 1 2 . 2. CU (u, v) exhibits tail dependence for a = 0, and tail independence for all a > 0.
3. The uniform family is star ordered with respect to a and b, therefore one can conclude that uniform Lorenz copulas are ordered according to the LTD (PK and PQD) dependence order.
4. The Gini index of the uniform family is GU = b−a 3(a+b) , which can never be equal to one. Therefore, by Proposition 4, the uniform Lorenz copula will never attain its upper Fréchet-Hoeffding bound M . (3) it is possible to obtain a closed form formula for the Kendall's τ associated to the uniform Lorenz copula.

Using Equation
Observe that, if a = 0, one has τU = 0 for every choice of b. But for a = 0 the uniform Lorenz copula necessarily exhibits tail dependence, hence this situation represents another pathological example of how measures of association should not be fully and acritically trusted when dealing with copulas [33,35].

Alchemies and Multiparametric Extensions
The class of Lorenz copulas is extremely rich and flexible. In the previous section a few examples were considered, starting from some well-known size distributions, but they only represent a small set of all the possible copulas one can actually generate. Just think about all the Lorenz curves currently available in the literature [4,10,25,39,48]. Besides flexibility, an appealing characteristic of the Lorenz approach to copulas is the possibility of importing into the Archimedean family many results developed in the study of inequality. A particularly interesting example is represented by the so-called "alchemy of Lorenz curves" discussed in [4,39]. The evocative term alchemy is used by Sarabia and Arnold to indicate a set of techniques for generating new Lorenz curves starting from given ones. Some relevant cases are presented in the following proposition, for the proof of which see [4].
Proposition 8. Let L1(p) and L2(p) be two Lorenz curves. Then, a new Lorenz curve L(p) can be obtained, for instance, via generator in Equation (30). By applying exponentiation and multiplication one can easily obtain the following family of three-parameter Lorenz generators with η ≥ 0, θ ∈ (0, 1) and γ ≥ 1. The related Lorenz curves have been studied extensively in [40]. From Equation (36) it is possible to obtain a three-parameter Lorenz copula, whose properties can be studied using the results of Section 2. For example, one can quickly find out that the copula obtained from L3P is almost never tail independent. This comes from the fact thatL 3P (1) = 0 for every choice of the parameters except for η = 0 and γ = 1.
By taking η = 0 in Equation (36), one obtains the mirrored Lorenz curve of a Sigh-Maddala random variable [45], which represents a pseudo-translation of a standard Pareto [25,36], so that the new variable has its lower bound shifted to zero. Because of this new lower bound the original Paretian tail independence is lost.
By setting γ = 1 θ , Equation (36) becomes the famous Genest and Ghoudi's generator [18] behind copula 4.2.15 in [35]. Thanks to the Lorenz approach, it is immediate to study the properties of the associated copula. First notice that the new generator corresponds to the mirrored Lorenz curve of a Lomax random variable [25]. The Lomax distribution has a lower bound at zero and no upper bound, hence the associated copula is absolutely continuous and never tail independent. Moreover, looking at the behavior of the density and noting that it is nowhere similar to the one of a log-normal distribution we can conclude, by using Theorem 3, that the copula will be tail dependent for every choice of the parameter (the Lomax is indeed known as a lognormal-like distribution [25]). Finally, by noting that the Lomax family is star ordered with respect to the parameter θ, the associated family of copulas is stochastically ordered, as noted by Genest and Ghoudi [18]. In particular, Proposition 6 guarantees that the family is LTD (PK and PQD) ordered. Table 1 summarizes and extends the results presented so far, listing some Lorenz generators and the properties of the related Lorenz copulas.

Conclusions
We have proposed an alternative approach to the generation of non-strict bivariate Archimedean copulas using the Lorenz curve, a powerful tool in the study of socio-economic inequality [12,13,25,48] and risk management [15,16,43,44]. The main advantages of the Lorenz-generation of copulas are threefold. First, the great number of Lorenz curves available in the literature allows for the generation of a large amount of copulas, which include existing cases, but also novelties. Second, every Lorenz copula can be easily characterized looking at some basic features of the non-negative finite-mean variable X underlying the Lorenz generator. In particular, we have shown that quantities like the Kendall's τ , or properties like upper tail dependence and stochastic dominance, can be inferred from X. Third, the possibility of importing into the world of copulas many of the results developed in the studies of inequality allows for many interesting considerations: from a novel perspective on the Gini index, as a measure of the distance of a Lorenz copula from its Fréchet-Hoeffding bounds, to the possibility of generating multiparametric copulas using some useful compositions rules for Lorenz curves.
Regarding the last point, it is interesting to notice that the opposite direction works as well. It is in fact possible to borrow tools from the theory of copulas and to apply them to the study of socio-economic inequality. Consider for example the Kendall's τ .
In terms of Lorenz curve, one has Now, observe that U (p) = p 0 µ F −1 (t) dt is an increasing function. Therefore one can rewrite (37) as Following [32], the quantity τ in Equation (38) is a valid inequality index with weighting function U . In particular, τ measures the distance between the Lorenz curve L(p) and the line of perfect equality LP E (p) = p, and in this it is similar to the Gini index. However, differently from the standard Gini, τ weights both L(p) and LP E (p) for the actual value of wealth, as represented by the quantile function F −1 (p). As a consequence, τ could be used for direct comparisons among countries, something not immediately possible using the Gini index, given its scale free nature [48].
To conclude, as far as future work is concerned, it would be interesting to investigate the possibility of extending the Lorenz approach to d-dimensional copulas, with d ≥ 3. Viable solutions could be the exploitation of nested constructions [35], or the use of multivariate Lorenz curves, as for example the Lorenz zonoid of [27]. For this second direction, however, one needs to remember that there exist more definitions of multivariate Lorenz curves [4,25], and there is no guarantee that they may all work for the purpose.  Table 1: Some examples of Lorenz copulas with their properties. Legend: TI = tail independence, AD = asymptotic tail dependence, TD = tail dependence, * = star order (which then implies the other orders as discussed in Section 1.3), − not available.
By Definition 2, X1 * X2 when L1(L −1 2 (x)) is convex. For a generic Lorenz curve L(p), associated to a non-negative random variable X with distribution function F and mean µ < ∞, one has L (p) = F −1 (p) µ . Now, assume that L1 and L2 are both twice differentiable. Then By the convexity of L2(L −1 1 (x)), the previous equation is equivalent to being non-decreasing, since µ1 and µ2 are always positive, and L −1 i (p), i = 1, 2, is a map from [0, 1] to itself.

Subsection 3.2 -Shifted Exponential Lorenz Copula
Consider the Lorenz curve of the shifted exponential distribution, i.e.
Notice that the maximum appearing in Equation (29) takes care of the fact thatL(u) +L(v) may be larger than 1.
Consider the quantile function of a Pareto random variable, F −1 (p) = xm(1 − p) − 1 α . Equation (39) becomes which is decreasing for every α1 < α2. Hence we can conclude that the shape parameter α orders Pareto random variables in the star sense. Proposition 6 then guarantees that the Pareto Lorenz copulas C1 and C2 associated to X1 and X2 are such that C1 LTD C2.