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Utilizing a unit Gompertz distorted copula to model dependence in anthropometric data
Communications for Statistical Applications and Methods 2023;30:467-483
Published online September 30, 2023
© 2023 Korean Statistical Society.

Fadal Abdullah Ali Aldhufairi1,a

aDepartment of Mathematics, College of Science, King Khalid University, KSA(Saudi Arabia)
Correspondence to: 1 Department of Mathematics, College of Science, King Khalid University, Abha 61421, KSA(Saudi Arabia). E-mail: aldhu1fa@cmich.edu
Received February 25, 2023; Revised April 9, 2023; Accepted June 5, 2023.
 Abstract
In this research, a conversion function and a distortion associated with the conversion function are defined and used to derive a unit power Gompertz distortion. A new family of copulas is built using the global distorted function. Four base copulas, namely Clayton, Gumbel, Frank, and Gaussian, are distorted into the family. Some properties including tail dependence coefficients and tail order are examined. Kendall’s tau formula is derived for new copulas when the base copula is Clayton, Gumbel, or Frank. The maximum pseudo-likelihood estimation method is employed, and a simulation study was performed. The log-likelihood and AIC are reported to compare the performance of the fitted copulas. According to the applied data, the results indicate that new distorted copulas with additional parameters improve the fit.
Keywords : distortion, copula, Gompertz, tail dependence, Kendall’s tau
1. Introduction

Dependence occurs between different events or co-movements in a variety of real-world situations, including finance, medicine, and insurance (McNeil et al., 2006; Cherubini et al., 2004; Klugman and Parsa, 1999). One of the functions commonly used to investigate this dependence is known as copula. Sklar (1959) explores the first notation for a copula, and he defines the copula as a multivariate distribution function with uniform margins; for further details, see Joe (2015). Dependence can be represented by one or more variables, and the choice of a fitted copula model for the dependence between random variables can be carried out independently from the selection of margins (Jaworski et al., 2010; Frees and Valdes, 1998).

Sklar’s theorem has been considered a main result for the copula function recognized in the literature, and from the result of this theorem, the copula method (or sometimes called the inversion method, e.g., see Nelsen, 2006), which plays a key role in deriving numerous copulas such as Clayton, Frank, Gumbel, and Gaussian from their distributions (Nelsen, 2006), is established. In recent work, the focus has been on generating new copulas with extra parameters to make copula models more flexible, thereby potentially providing a better fit (Aldhufairi and Sepanski, 2020; Xie et al., 2019).

The motivation of this study is to highlight an alternative approach for deriving a new distortion, generalizing some results of distorted copulas, and constructing a new family of distorted copulas with three parameters via a unit power Gompertz distortion, and that family can be utilized to model several phenomena, such as in financial studies. This is robust, as new copula models indicate a better fit for real data.

The remainder of this paper is organized as follows: In Section 2, the family of the UPG copulas and their corresponding conditional copula and copula density are formulated, and some examples are offered after limiting cases in parameters are provided. Section 3 presents the conversion function and the unit power Gompertz (UPG) distortion, and it studies the admissibility conditions on the parameters. Section 4 examines tail behaviors, and the Kendall’s tau measure is detailed in Section 5. To assess the performance of the new UPG-distorted copula models, we apply the data set in Section 6, while the concluding remarks are provided in Section 7.

2. Copula models

Let X1, X2, . . ., Xn be continuous random variables. Consider Fi(xi) = P(Xixi) corresponds to a univariate margin. Let H be a multivariate distribution function.

2.1. Copula

Two properties hold for any base copula C: 1) C is grounded, that is, C(u1, . . ., un) = 0 if ui = 0 for at least one index i ∈ {1, 2, . . ., n} and C(1, . . ., 1, ui, 1, . . ., 1) = ui if all the coordinates of (u1, . . ., ui, . . ., un) are 1 except ui; and 2) C is n–increasing, that is, for all (a1, . . ., an), (b1, . . ., bn) ∈ [0, 1] × · · · × [0, 1] (n times) such that aibi for all 1 ≤ in, we have Δ(an,bn)(n)Δ(an-1,bn-1)(n-1)Δ(a1,b1)(1)C(u1,u2,,un)0, where Δ(ai,bi)(i)C(u1,u2,,un)=C(u1,,ui-1,bi,ui+1,,un)-C(u1,,ui-1,ai,ui+1,,un). From Sklar’s theorem (see Nelsen, 2006), there exists a unique copula C such that

H(x1,x2,,xn)=C(F1(x1),,Fn(xn)).

If Ui=Fi(Xi)iidUnif[0,1], for i = 1, 2, . . ., n, we can write

C(u1,u2,,un)=H(F1-1(u1),,Fn-1(un)).

If H is n times differentiable function, the joint probability density function h can be obtained by the following

h(x1,x2,,xn)=nH(x1,x2,,xn)/xnx2x1.

The conditional copula and copula density can be, respectively, formulated by C(u1, . . ., ui–1, ui+1, . . ., un|ui) = ∂C(u1, . . ., ui–1, ui, ui+1, . . ., un)/∂ui and c(u1, u2, . . ., un) = ∂nC(u1, u2, . . ., un)/∂un · · · ∂u2u1.

2.2. Distortion

A distortion T, which maps from [0, 1] to [0, 1], is defined as an increasing and continuous function satisfying T(0) = 0 and T(1) = 1. It is described as the simultaneous distortion of the margins and copula in Valdez and Xiao’s (2011) work, see Durante and Sempi (2005). For i = 1, 2, 3, . . ., n, if the function Ti: [0, 1] → [0, 1] is increasing and continuous with Ti(0) = 0 and Ti(1) = 1, Di Bernardino and Rulliere (2013) state that the global distorted distribution function of H is defined as

H˜(x1,x2,,xn)=TC(T1-1(F1(x1)),,Tn-1(Fn(xn))).

The marginal distribution of is given by F˜i=TTi-1Fi, for i = 1, 2, . . ., n. Charpentier (2008) and Valdez and Xiao (2011) dealt with a particular situation where T = T1 = · · · = Ti, and their study involved the bivariate case.

Let ℝ be a set of real numbers. Let ζ: ℝ → ℝ be any bijective and increasing function, then ζ is said to be a conversion function, and by Di Bernardino and Rulliere (2013), the associate distortion to ζ is given by Tζ: [0, 1] → [0, 1] such that

Tζ(u)=logit-1(ζ(logit(u)))=eζ(log(u/(1-u)))1+eζ(log(u/(1-u))).

For all u ∈ [0, 1], Tζ is increasing because Tζ(u)0, and Tζ(0) = 0, Tζ(1) = 1. Furthermore, the inverse function of Tζ satisfies Tζ-1=Tζ-1.

Denote the distorted function by , which does not have necessarily to be a copula, such that (x1, x2, . . ., xn) = (1(x1), . . ., n(xn)). According to Proposition 2.5, as derived from the work of Di Bernardino and Rulliere (2013), the function is only affected by the external distortion T, that is,

C˜(u1,,un)=T(C(T-1(u1),,T-1(un))).

T is deemed to be an admissible distortion if (2.3) is a copula. Given that s1, s2, r1, and r2 are real numbers with two intervals I1 and I2 in ℝ such that r1r2 and s1s2, it is said that the function L: I1 × I2 → ℝ is totally positive of order 2, denoted by TP2, if

L(r1,s1)L(r2,s2)L(r1,s2)L(r2,s1).

In general, if the base bivariate copula C is TP2 and T ○ exp is log-convex, then the bivariate distorted function is TP2, and thus, it satisfies the 2-increasing property based on Lemma 3.1 stated by Durante et al. (2010). In addition to the aforementioned property, because C is a base copula, then the bivariate distorted function is a copula because the grounded property is preserved under the distortion T.

3. Conversion function and distortion

From (2.2), set w = logit(u) = log(u/(1 – u)). For u ∈ [0, 1], we have w ∈ [−∞,∞], and the inverse transform is u = ew/(1 + ew).

3.1. Examples of distortions

Here, in the following examples, we attempt to define the conversion function ζ such that the associate distortion in (2.2) satisfies the definition of distortion and is convex. The associate distortion functions we derive below are found in the work of Sepanski (2020).

  • - If ζ1(u) = − log (e(− log(eu/(1+eu)))δ1 −1), δ1 ≤ 1, the associate distortion Tζ1 to ζ1 produces the Weibull-log distortion given by Tζ1 (u) = e−(− log u)δ1, where the inverse is given by Tζ1-1(u)=e-(-log u)1/δ1.

  • - If ζ2(u) = − log (1/(1 – (1 + eu)δ1) – 1), δ2 ≤ 1, the associate distortion Tζ2 to ζ2 produces the dual-power distortion given by Tζ2 (u) = 1–(1–u)δ2, where the inverse is given by Tζ2-1(u)=1-(1-u)1/δ2.

  • - If ζ3(u) = − log ((eu/(1 + eu))δ3 – 1), δ3 ≥ 1, the associate distortion Tζ3 to ζ3 produces the power distortion given by Tζ3 (u) = uδ3, where the inverse is given by Tζ3-1(u)=u1/δ3.

  • - If ζ4(u) = − log ([−δ4/ log((1 + eδ4+u)/(1 + eu))] – 1), δ4 > 0, the associate distortion Tζ4 to ζ4 produces the logarithmic distortion given by Tζ4 (u) = − log(1 – u(1 – eδ4 ))/δ4, where the inverse is given by Tζ4-1(u)=(1-e-δ4u)/(1-e-δ4).

  • - If ζ5(u) = − log ([1 – log (eu/(1 + eu))]δ5 – 1), δ5 > 0, the associate distortion Tζ5 to ζ produces the Lomax-log distortion given by Tζ5 = (1–log u)δ5, where the inverse is given by Tζ5-1(u)=e(1-u-1/δ5).

Note that the functions ζ1, ζ2, ζ3, ζ4, and ζ5 are bijective and increasing, thereby satisfying ζ(−∞) = −∞ and ζ(∞) = ∞. If δ1 = δ2 = δ3 = 1, from (2.1),

C(Tζ1-1(u1),Tζ2-1(u2),Tζ3-1(u3))=C(u1,u2,u3).

3.2. Gompertz distortion

The Gompertz distribution was derived by Gompertz (1825), and it has several real applications; see Ahuja and Nash (1967) and references therein. It is one of the distributions mentioned by Durante et al. (2010), and hence, we intend to use this distribution to construct our new copula models.

Define

ζ(v)=-log (11-G(log (1-log (ev/(1+ev))))-1),

where G is a distribution function. If ζ0(v) = 1 −G(log (1 – log (ev/(1 + ev)))), then the first derivative of ζ0 with respect to v is given by ζ0(v)=g(log (1-log(ev/(1+ev))))/[(1+ev)(1-log(ev/(1+ev)))], where G’ = g. Thus, ζ(v)=ζ0(v)/[(1-ζ0(v))ζ0(v)]. ζ’ is increasing. Let G be a power Gompertz (PG) distribution defined as G(z) = 1 – exp[−ba−1(eaz – 1)], z > 0, for a, b > 0. Applying (3.2), we obtain ζPG(v) = − log (eb[(1–log (ev/(1+ev)))a −1]/a −1). As a result, from (2.2), the associate UPG distortion to ζPG is

TζPG(v)=e-b[(1-log v)a-1]a.

The probability density function and quantile function of V are defined as

tζPG(v)=TζPG(v)=bv(1-log v)a-1e-b[(1-log v)a-1]a,

and

TζPG-1(v)=e[(1-(1-(alog v)b)1a],

respectively.

When a = 1 in (3.3), the UPG distortion is transformed into the power distortion, which has the form TζPG (u) = ub, b > 0. The power distortion is a distortion function because it is increasing on [0, 1] with T(0) = 0 and T(1) = 1 for all b > 0.

For any base copula C, the family of the UPG distorted distributional function (shortly, the UPG distorted function) is overall given by

C˜TζPG(u1,,un)=exp (-ba-1[(1-log C(e[1-(1-(alog u1)/b)1/a],,e[1-(1-(alog un)/b)1/a]))a-1]).

If a = 1 and b = 1, the UPG distorted function TζPG becomes equal to a base copula C, and thus, TζPG is a copula function.

From (2.1) and (3.1), the following distorted function

C˜T123(u1,u2,u3)=TζPGC(Tζ1-1(u1),Tζ2-1(u2),Tζ3-1(u3))

is equal to a base copula C(u1, u2, u3) when a = b = δ1 = δ2 = δ3 = 1. It is worth to notice, that C˜T123(u1,u2,u3)=C(Tζ1-1(u1),Tζ2-1(u2),Tζ3-1(u3)) when a = b = 1.

General bivariate forms of the conditional cdf and copula pdf of can be found in Aldhufairi et al. (2020).

According to Theorem 3.2, as derived by Durante et al. (2010), we show that TζPG ○ exp: (−∞, 0] → [0, 1] is log-convex under the admissibility mentioned in the following corollary.

Corollary 3.1. Let TζPG(u) be the UPG distortion for u ∈ [0, 1]. TζPG is a log-convex function if 0 < a ≤ 1 and b > 0.

Proof: Define B(x) = log ○ TζPG ○ exp(x). Then, the function B(x) = −b[(1 – x)a – 1]/a has its first and second derivatives with respect to x as follows: B’ (x) = b(1 – x)a–1 and B” (x) = b(1 – a)(1 – x)a–2. Thus, the second derivative B” is non-negative if 0 < a ≤ 1 and b > 0 for x ∈ (−∞, 0].

3.3. Examples of distorted copulas

We focus on the bivariate case in the examples of proposed copula models offered in this section. According to Joe (1997), the base copulas are TP2 in any of the following examples, which are constructed using (3.3), (3.4), and (2.3).

One popular class of copulas is Archimedean, defined as if a generator function φ: [0, 1] → [1,∞) is continuous, strictly decreasing, and convex with φ(1) = 0 exists and generates the copula via C(u1, . . ., un) = φ[−1](φ(u1) + · · · + φ(un)). If φ(0) = ∞, then φ[−1] = φ−1. If a base copula C with a generator φ belongs to the Archimedean class, then the distorted copula does too. Its distorted generator is given by ϕ̃ = φT−1, as demonstrated by Aldhufairi et al. (2020). One can then rewrite (3.5) as follows:

C˜TζPG(u1,,un)=φ˜-1(φ˜(u1)+φ˜(u2)++φ˜(un)),         φ˜(u)=φ(e[1-(1-(alog v)/b)1/a]).

Di Bernardino and Rulliere (2013) report on page 7 that T is an admissible distortion if and only if ϕ̃ is a n–monotone function. This result allows for the generalization of the distorted copulas. It can be accomplished by expanding the findings of Theorem 3.2 and Lemma 3.1 from the work of Durante et al. (2010). Furthermore, there is a need to carefully investigate which of the copulas are multivariate and totally positive of order 2. This may create a gap for future research.

Another popular class of copulas is referred to as extreme-value, defined as if a convex function A: [0, 1] → [1/2, 1], satisfying A(0) = A(1) = 1, and max{t, 1 – t} ≤ A(t) ≤ 1 exists and produces the copula via C(u, v) = exp[log(uv)A(log(v)/ log(uv))], u, v ∈ [0, 1] (Gudendorf and Segers, 2010).

Unlike Clayton and Frank that belong to the Archimedean class only, Gumbel belongs to the Archimedean and extreme-value classes. Some copulas have no closed form; for instance, the Gaussian copula that is proposed from the Gaussian distribution.

Example 3.1. (UPG-Clayton copula) The Clayton copula with its generator is given by

C(u1,u2;θ)=(u1-θ+u2-θ-1)-1θ,         θ>0,φ(u)=(u-θ-1)θ.

The UPG-distorted Clayton copula (shortly, UPG-Clayton copula) can be expressed as

CTζPG(u1,u2)=exp (-ba[(1+θ-1log (i=12e-θ[1-(1-ab-1log ui)1a]-1))a-1]),

and its generator is ϕ̃(u) = {eθ[1–(1–ab−1 log u)1/a] – 1}/θ.

Example 3.2. (UPG-Gumbel copula) The Gumbel copula with its generator is given by

C(u1,u2;θ)=exp {-[(-log u1)θ+(-log u2)θ]1θ},         θ1,φ(t)=(-log t)θ.

The UPG-distorted Gumbel copula (shortly, UPG-Gumbel copula) is

CTζPG(u1,u2)=exp (-ba[(1+(i=12((1-ab-1log ui)1a-1)θ)1θ)a-1]),

and its generator is ϕ̃(u) = [(1 – ab−1 log u)1/a – 1]θ.

Example 3.3. (UPG-Frank copula) The Frank copula with its generator is given by

C(u1,u2;θ)=-θ-1log (1+[(e-θu1-1)(e-θu2-1)]e-θ-1),         θ0,         φ(u)=-log (e-θu-1e-θ-1).

The UPG-distorted Frank copula (shortly, UPG-Frank copula) is

CTζPG(u1,u2)=exp (-ba[(1-log (-θ-1log (1+i=12(e-θe[1-(1-ab-1log ui)1/a]-1)e-θ-1)))a-1]),

and its generator is ϕ̃(u) = − log[(eθe1–(1–ab–1 log u)1/a – 1)/(eθ – 1)].

Example 3.4. (UPG-Gaussian copula) The Gaussian copula is given by

C(u1,u2;θ)=Φ2(Φ-1(u1),Φ-1(u2))

with its parameter θ ∈ [0, 1], where Φ−1(s) is the quantile function of the univariate standard Gaussian distribution

Φ(s)=-s12πe-(x2/2)dx.

The bivariate standard Gaussian distribution function Φ2 is given by

Φ2(s1,s2)=-s1-s212π1-θ2exp (-x2-2θxy+y221-θ2)dxdy.

Thus, the UPG-distorted Gaussian copula (shortly, UPG-Gaussian copula) is a direct application of this expression TζPG(C(TζPG-1(u1),TζPG-1(u2))). That is,

CTζPG(u1,u2)=TζPG[Φ2(Φ-1(TζPG-1(u1)),Φ-1(TζPG-1(u2)))].

3.4. Limiting case

The following proposition investigates the limit of the UPG copulas from a given copula C when one or both parameters go to a boundary.

Proposition 3.1. Consider the UPG-distorted function in (3.5), where 0 < a ≤ 1 and b > 0. Then, for any base copula C, if this exponent of b(1-log C(TζPG-1(u1),,TζPG-1(un)))a-1 goes to log u1 + · · · + log un whenever b→∞, we have that CTζPG approaches the independence copula.

Proof: Let xi = e[1–(1–ab–1 log ui)1/a], for i = 1, . . ., n. Set m = 1/b and consider Im = C(x1, . . ., xn). As C is a base copula, Im → 1 as m → 0. By chain rule, the derivative of Im with respect to m is given by

dImdm=C(x1,,xn)x1dx1dm++C(x1,,xn)xndxndm.

For i = 1, 2, . . ., n, we have

dxidm=(log ui)(1-amlog ui)1a-1exp [1-(1-amlog ui)1a].

The limit of the exponent term in (3.5) as b→∞exists, by L’Hopital’s rule,

limm0(1-log Im)a-1am=-limm0(1-log Im)a-1ImdImdm.

As m → 0, we have dxi/dm → log ui, for i = 1, 2, . . ., n. Thus, we obtain the following limb→∞CTζPG (u1, . . ., un) = u1 · · · un, which is the independent copula.

The limit of the UPG-distorted copula in the parameter obtained from a base copula C can be calculated by finding the limit of the base copula in the parameter. The limits of the chosen base copulas were evaluated in Joe (2015). For example, as θ → 0+, the bivariate Clayton copula C approaches the independent copula. As a result, we derive the following: C(T−1(u1), T−1(u2)) → T−1(u1)T−1(u2), as θ → 0+. Thus, the distorted bivariate copula TζPG approaches the copula exp(-ba-1[((1-i=12(alog ui)/b)1/a-1)a-1])

whenever θ → 0+.

3.5. Frèchet bounds and independent case

Here, we look at three common cases that show the impact of choosing the initial copula C in .

  • - AssumeC is counter-monotonic, which is the lower Frèchet, in bivariate case n = 2. IfC(T−1(u1), T−1 (u2)) = max{T−1(u1) + T−1(u2) – 1, 0}, then we have (u1, u2) = TC(T−1(u1), T−1(u2)) = max{T(T−1(u1)+T−1(u2)–1), 0} since T is monotonically increasing. For example, when TζPG (u) = ub, with b ≥ 1, then C˜(u1,u2)=max{(u1-b1+u2-b1-1)-1/b1,0}, where b1 = −1/b ∈ [−1, 0), which is the Clayton copula.

  • - If C is comonotonic, which is upper Frèchet, given by C(u1, . . ., un) = min(u1, . . ., un), then because TζPG increases, we have C˜(u1,,un)=TζPG(min(TζPG-1(u1),,TζPG-1(un)))=min(u1,,un). Thus, is also comonotonic. In this case, we can see that TPG has no effect on the result of the base copula C. This implies that TPG plays the role of an identity distorter, T(u) = u.

  • - Consider that C is an independent copula. If C(T−1(u1), . . ., T−1(un)) = T−1(u1)× · · · ×T−1(un), then (u1, . . ., un) = TC(T−1(u1), . . ., T−1(un)) = T(T−1(u1) · · · T−1(un)). For example, if TζPG (u) = ub, b > 0, we have (u1, . . ., un) = u1 · · · un. Thus, is the independent copula.

4. Tail orders and dependence coefficients

This section explores the lower and upper tail behaviors of the UPG copulas from a given base copula C. One possible thought beyond the addition of new parameters is to produce new models of copulas that adapt to the different behaviors of tail dependence to estimate risky and extreme events.

The survival copula is defined as Ĉ (u1, u2, . . ., un) = P(1 – U1u1, 1 – U2u2, . . ., 1 – Unun) = (1 – u1, 1 – u2, . . ., 1 – un), where is the joint survival function of C. The survival bivariate copula can be written as Ĉ(u1, u2) = u1 +u2 –1+C(1–u1, 1–u2). A regularly varying function with index ξ is defined by limu→0+ (γu)/(u) = γξ for all γ > 0. If ξ = 0, is said to be slowly varying, see Hua and Joe (2013). If a lower tail dependence coefficient (λL) of a base copula C exists, the lower tail dependence coefficient (λT,L) of the distorted copula CT is given by

λT,L=limu0+T(C(T-1(u),T-1(u)))u=limu0+T(C(u,u))T(u).

If an upper tail dependence coefficient (λU) of C exists, the upper tail dependence coefficient (λT,U) of CT is given as follows:

λT,U=2-limu1-1-T(C(T-1(u),T-1(u)))1-u=2-limu0+1-T(C(u,u))1-T(u).

If we consider the following expansions

(1+u)a~1+au,         log(1-u)~-u,   eu~1+u,         as u0,

we obtain TζPG-1(1-u)~1-u/b, as u → 0+. For C(u, v) ~ uκL(u), the UPG-distorted copula has a lower tail order of κL because

TζPG(C(TζPG-1(u),TζPG-1(v)))=eb[(1-log C(TζPG-1(u),TζPG-1(v)))a-1]/a~eb[(1-alog(TζPG-1(u))κL(TζPG-1(u)))-1]/a~eb[κL(1-(1-ab-1log u)1/a)+log (TζPG-1(u))]~eb[kLb-1log u+log (TζPG-1(u))]~uκL[(TζPG-1(u))]b.

Proposition 4.1. Let and * be two slowly varying functions. As u → 0+, assume C(u, u) ~ uκL(u) and (1 – u, 1 – u) ~ uκU(u) at 0+. The UPG copula CTζPG satisfies the following: κLTζPG = κL, λTζPG,L = (λL)b if a = 1, κTζPG, U = κU, and λTζPG,U = λU.

Proof: The lower tail dependence coefficient of CTζPG is given by

λTζPG,L=limu0+TζPG(C(u,u))TζPG(u)=limu0+e-b[(1-log C(u,u))a-1]/ae-b[(1-log u)a-1]/a=limu0+e-ba-1(1-log u)a[(1-log C(u,u)1-log u)a-1].

We can see some possibilities for (4.1). If a < 1, then λTζPG,L = 0, which means there is no lower tail dependence. If a = 1, then λTζPG,L = limu→0+eb[log (u/C(u,u))] = (λL)b. However, the following can be obtained from (3.5) as

[1-log C(TζPG-1(1-u),TζPG-1(1-u))]a~[1-log C(1-u/b,1-u/b)]a~[1-log (1-2u/b+C¯(1-u/b,1-u/b))]a~[1-log (1-2u/b+(u/b)κU*(u/b))]a~[1+2u/b-(u/b)κU*(u/b)]a~[1+a(2u/b-(u/b)κU*(u/b))]=KPG.

It follows that

C^TξPG(u,v)=2u-1+e-ba-1[(1-log C(TζPG-1(1-u),TζPG-1(1-u)))a-1]~2u-1+1-ba-1(KPG-1)~2u-ba-1[2au/b-a(u/b)κU*(u/b)]~b1-κUuκU*(u/b).

As a result, the upper tail order κTζPG,U of CTζPG is κU. The upper tail dependence coefficient of CTζPG is given by

λTξPG,U=2-limu1-1-e-b[(1-log C(u,u))a-1]/a1-e-b[(1-log u)a-1]/a=2-limu1-{e-b[(1-log C(u,u))a-1]/ae-b[(1-log u)a-1]/a[1-log C(u,u)1-log u]a-1uC(u,u)dC(u,u)du}=λU.

Table 1 provides a summary of tail order and dependence for the base and new copulas. The new UPG-distorted copulas can accommodate extra parameters in the lower tail dependence when the base copula has the lower tail dependence.

Figure 1 displays contour plots of a bivariate pdf, h, for three new copulas distorted from Clayton, Gaussian, and Frank. The parameters in all base copulas have been chosen when Kendall’s tau has a value of 1/2.

A bivariate copula C is said to be symmetric when C(u1, u2) = C(u2, u1) for u1, u2 ∈ [0, 1]. As shown in Figure 1, the resulting new copulas from the given symmetric Frank and Gaussian copulas are asymmetric when a or b differ from a value of 1.

5. Kendall’s tau

The section derives the Kendall’s tau formulas for three copulas, namely, UPG-Clayton, UPG-Gumbel, and UPG-Frank. Then, it studies the ordering of concordance based on the formulas we derive.

Copulas offer a natural approach for measuring the dependence between two random variables, and one of these measures is called Kendall’s tau, which is a non-parametric measure.

The general and Archimedean formulas of Kendall’s tau for the bivariate distortion T can be found in Aldhufairi et al. (2020).

Note that the values computed from the Kendall’s tau formula can either be an increase or decrease in one parameter while the remaining parameters are held constant. If the values obtained from the Kendall’s tau formula increase but never decrease, or decrease but never increase, in such a parameter r, then the function is ordered by r.

Example 5.1. Let φ be a generator function for Clayton. The following can be formulated as φ(u)/φ’(u) = −θ−1u(1 – uθ). Kendall’s tau of the UPG-Clayton is given by

τφTζPG=1-4b2θ01(1-uθ)u(1-log u)2a-2e-2b((1-log u)a-1)/adu.

The UPG-Clayton copula is ordered in a and b because checking the first derivative with respect to a and b reveals that the UPG-Clayton copula is increasing in a and b.

Example 5.2. Let φ be a generator function for Gumbel. Then, φ(u)/φ’(u) = −u(− log(u))/θ, and thus, Kendall’s tau for the UPG-Gumbel copula is given by

τφTζPG=1-4b2θ01(-log u)u(1-log u)2a-2e-2b[(1-log u)a-1]/adu.

The UPG-Gumbel copula is also ordered in a and b because it is increasing in a and b.

Example 5.3. Let φ be a generator function for Frank. Then, φ(u)/φ’ (u) = −θ−1(1 – eθu) log [(eθu – 1)/(eθ – 1)]. Thus, Kendall’s tau of the UPG-Frank with setting v = θu is given by

τφTζPG=1-4b20θ(1-ev)(1-log (v/θ))2a-2v2e2b((1-log(v/θ))a-1)/alog [e-v-1e-θ-1]dv.

As shown in Figure 2, the concordance ordering in a or b can fail to hold. Nonmonotonic curves can be seen with parameters a and b in the plots titled with fixed θ = 15 and b = 3 for the UPG-Frank copula.

6. Application

Here, the data set is analyzed utilizing the R programming language to assess the performance of the new UPG-distorted copula models in this section. Based on the chosen base copulas, namely, Clayton, Gumbel, Frank, and Gaussian, the Akaike’s information criterion (AIC) statistics are used to determine the best copula model. We perform the Cramer-von Mises (CvM) goodness-of-fit test (Genest et al., 2009) and compare the performance of the statistics with the base copulas, where the CvM test statistics measure the sum of square deviations between the empirical cdf and an estimated copula cdf. Larger CvM values are less desired. The null hypothesis of the CvM is that a candidate copula models bivariate data.

The data are analyzed using the maximum pseudo-likelihood estimation (MPLE), see Joe (2015) for the bivariate case. The MPLE maximizes

i=1mlog [cT(u1,i,u2,i,,un,i;θ,a,b)],

where ui, j = Fi(xi, j), i = 1, 2, . . ., n, j = 1, 2, . . ., m, are the pseudo-observations (shortly, pseudo-obs) and cT is the copula pdf.

6.1. A simulation study

In this subsection, a simulation study is carried out to examine the flexibility of the new copulas as a result of unit Gompertz distortion. For simulating bivariate data from copulas, see Aldhufairi et al. (2020) for details about the conditional method they describe.

First, we simulated four bivariate data sets from Clayton, Gumbel, Frank, and Gaussian copulas, each with 1500 observations, using parameter values of 2, 2, 5.82, and 0.71, respectively. Then, using UPG-Clayton, UPG-Gumbel, UPG-Frank, and UPG-Gaussian copulas with parameter values of (2, 0.5, 1.25), (2, .5, 2), (0.71, 0.25, 2), and (5.82, 0.5, 1.5), four data sets of 1500 each were respectively produced. The parameters are estimated using the pseudo-likelihood estimation method in (6.1). Following that, each data set produced from the UPG-distorted copulas was fitted to using the base copula models, and vice versa.

In the first and second rows of the P-P plots of the empirical cdf and the estimated cdf in Figure 3, the black curve is from the true copula model, while the curves that are not black are from fitting the estimated copula models. We can identify which copula model has the worst fits by looking at the departure curves from the black curve. For instance, it appears that the Clayton copula has the worst models since the red curve strongly deviates from the black curve. Compared to the second row of data, the first row of data generated from the base copulas appears to be well approximated by various UPG-distorted copula models. It would assume that UPG-distorted copulas with additional parameters will be more adaptable and enhance fit.

6.2. Data of anthropometric measurements

In this subsection, we use the data of 783 observations for anthropometric measurements on four variables, body mass index (BMI), body adiposity index (BAI), body fat percentage (BFP), and widest circumference (WC). A source can be assessed at the website: https://figshare.com/articles/dataset. The summary statistics for the four measurements are presented in Table 2. Data were gathered from 437 women and 346 men between the ages 25 and 80 years, and their average age was approximately 49 years.

The widest circumference was in centimeters, and it has been converted to meters. Age is in years, and BMI is in kg/m2. We calculated the BFP measurement as BFP = (1.39*BMI) + (0.16*Age) – (10.34*S) −9, where S = 0 for men and S = 1 for women, and the BAI measurement was based on a formula placed on the website: https://www.omnicalculator.com/health/bai. The four variables consider the risk to human health, as mentioned by Sapporo and Gongs in (2020), who carry out similar work.

The scatter plots in Figure 4 show how anthropocentric variables are related. Here, the Kendall’s tau values of the samples among each of the two selected variables are shown in Figure 4, as well. The correlation matrix, for instance, shows that the highest tau value is 0.51 between BAI and WC and the lowest tau value is 0.25 between BFP and WC.

Figure 5 gives an important sign for any lower or upper tail dependence that may help in assigning a suitable copula. There appear to be upper and lower tail dependencies, as demonstrated in Figure 5, and as a result, the UPG-distorted copula models look appropriate due to additional parameters and may be flexible to improve the fit. To illustrate more, the UPG-Gaussian copula model will perform better than the Clayton, Gumbel, and Frank copula models if there is upper and lower tail dependence between BAI and WC.

The results of the distorted copula models are presented in Table 3, which include MPLE, AIC, θ̂, â, and . Here, we simulate 783 Clayton, Gumbel, Frank and Gaussian observations, and then, we calculate the CvM test based on 1500 replicates. The Clayton copula model was not suitable for all situations in Table 3, so its performance is the worst. Gumbel, Frank, and Gaussian provide a good fit in terms of MPLE and AIC. The UPG-distorted copula models improve the fit in terms of MPLE and AIC. They have the ability to improve parameter estimations; note that all standard errors are small.

Based on the applied data, though the UL-distorted models may not perform as well as the UPG-distorted models, the UL-distorted copulas are anticipated to improve the model fit in terms of MPLE and AIC more than the base copula models.

The new distorted model of the Clayton copula successfully fits the data for all the dependent situations in Table 3, though UPG-Gumbel, UPG-Frank, and UPG-Gaussian continue to maintain their best overall performance model positions. As shown in Table 4, no distorted copula model would be rejected, as far as the CvM statistical test and its p-values are concerned. Table 5 summarizes the best copula models for each situation among two anthropometric measurements. Additionally, the values of Kendall’s tau and coefficients of the upper tail dependence are calculated and reported in Table 5. Furthermore, the values and coefficients between the base and distorted models are close to each other, and they reflect the values of Kendall’s tau shown in Figure 4.

7. Concluding remarks

Herein, we define the associate distortion that is linked to the conversion function, and then, it is used to construct a UPG distortion with two parameters. Additionally, a global distorted distribution function is used to construct a new family of the UPG-distorted function. New distorted bivariate copulas, namely, UPG-Gumbel, UPG-Clayton, UPG-Frank, and UPG-Gaussian, are explicitly expressed and given on the basis that the base copulas are, respectively, Gumbel, Clayton, Frank, and Gaussian. We look at the effect of the countermonotonic or comonotonic copulas on the distorted copula. For any base copula, the limiting cases in parameters for the family of the UPG-distorted copulas are carefully examined. The tail behaviors are investigated for the UPG-distorted copula. We derive Kendall’s tau formulas for UPG-Clayton, UPG-Gumbel, and UPG-Frank. Furthermore, these formulas are used to measure dependence in proposed copula models and examine the order of concordance. Based on the results of the application, the new distorted bivariate copula models are the best overall relative to their corresponding base bivariate copula models. In future research, we will attempt to extend our base and distorted copula models in the Application section to a high- or 4-dimensional base, and then, we can compare their performances. This paper would allow for a reasonable extension of this study to n-dimensional distorted copulas.

Figures
Fig. 1. Density contour plots with standard normal margins. The parameter is chosen so that the base copula has Kendall셲 tau of 1/2. The first and second columns show contour plots of the base and UPG-distorted copulas with (, a, b) = (2, 1, 1), (0.71, 1, 1), and (5.82, 1, 1). The contour plots for the UPG-Clayton (2, 0.5, 1.25), UPGGaussian (0.71, 0.25, 2), and UPG-Frank (5.82, 0.5, 1.5) are shown in the third column.
Fig. 2. Kendall셲 tau surface plot for the UPG-Frank copula, which displays tau values at fixed one of the parameters = 15, a = 0.5, or b = 3 for the other two parameters.
Fig. 3. P-P plots show the estimated and empirical cumulative distributions for different theoretical models. Results from fitting data obtained from the Clayton, Gumbel, Frank, and Gaussian copulas are shown in the first row. Data fitting results generated using the UPG-Clayton, UPG-Gumbel, UPG-Frank, and UPG-Gaussian copulas place in the second row.
Fig. 4. The correction matrix. Scatter plots of Kendall셲 tau correlation between anthropometric measurements (BMI, BAI, BFP, and WC) with a fitted line. Distribution of each variable in the data is diagonal in the matrix.
Fig. 5. Scatter plots between pseudo anthropometric measurements observations (pseudo-obs (BMI), pseudo-obs (BAI), pseudo-obs (BFP), and pseudo-obs (WC)).
TABLES

Table 1

Tail orders and dependence coefficients for copulas

Initial CopulaκL or λLκU or λU
GumbelκL = 21/θλU = 2 – 21/θ
ClaytonλL = 2−1/θκU = 2
FrankκL = 2κU = 2
GaussianκL = 2/(1 + θ)κU = 2/(1 + θ)

UPG-CopulaκLTζPG or λLTζPGκUTζPG or λUTζPG

UPG-GumbelκLTζPG = 21/θλU = 2 – 21/θ
UPG-ClaytonλLTζPG = 2b/θ if a = 1κUTζPG = 2
UPG-FrankκLTζPG = 2κUTζPG = 2
UPG-GaussianκLTζPG = 2/(1 + θ)κUTζPG = 2/(1 + θ)

Table 2

Summary statistics of BMI, BAI, BFP, and WC variables

BMIBAIBFPWC
Minimum15.73−6.6810.830.24
Maximum37.2938.1347.841.09
Median23.2422.0227.250.81
1st quartile21.1518.7121.820.75
3st quartile25.7225.1932.160.88
Standard deviation3.335.267.190.10

Table 3

MPLE, AIC, θ̂ â, , and parameter estimates (standard errors) for the UPG-distorted (base) and UL-distorted copula models

FamilyMPLEAICθ̂â
BMI & BAIUPG-Clayton220.80(139.20)−435.59(−276.33)0.017(0.002)(2.092(0.084))0.713(0.002)0.068(0.001)
UPG-Gumbel271.08(256.40)−536.17(−510.82)1.399(0.001)(1.919(0.053))0.518(0.001)0.190(0.001)
UPG-Frank259.64(257.70)−513.29(−513.42)2.824(0.284)(5.920(0.308))0.306(0.041)0.441(0.041)
UPG-Gaussian260.26(258.70 )−514.52(−515.35)0.831(0.051)(0.699(0.012))0.991(1.023)5.756(0.237)
UL-Clayton205.25−404.512.845(0.630)2.702(0.252)0.550(0.021)
UL-Gumbel270.67−535.331.569(0.065)1.001(0.104)4.236(1.211)
UL-Frank236.33−466.664.993(0.308)1.001(0.046)2.721(0.092)
UL-Gaussian259.96−513.930.784(0.012)2.717(0.126)1.005(0.001)

BMI & BFPUPG-Clayton170.81( 77.19)−335.61(−152.37)0.001(0.003)(1.717(0.066) )0.437(0.262)0.261(0.262)
UPG-Gumbel226.75(224.90)−447.50(−447.80)1.668(0.070)( 1.762(0.079))0.114(1.447)6.825(2.798)
UPG-Frank206.57(177.20)−407.14(−352.44)2.609(0.327)(4.411(0.378))0.505(0.002)0.184(0.002)
UPG-Gaussian251.16(227.30)−496.31(−452.64)0.417(0.014)(0.668(0.025))0.998(0.001)0.023(0.001)
UL-Clayton137.18−268.362.536(0.291)2.730(1.072)1.172(0.002)
UL-Gumbel226.77−447.551.661(0.075)4.913(1.632)3.547(1.025)
UL-Frank164.36−322.733.381(0.277)1.001(0.030)2.720(0.063)
UL-Gaussian227.84−449.732.758(0.014)2.717(0.103)1.003(0.006)

BMI & WCUPG-Clayton241.48(116.70)−476.96(−231.35)0.007(0.001)(2.271(0.092))0.412(0.002)0.059(0.001)
UPG-Gumbel293.13(285.40)−580.25(−568.86)1.592(0.047)(2.019(0.052))0.669(0.001)0.102(0.001)
UPG-Frank281.61(273.80)−557.22(−545.59)3.489(0.308)(6.271(0.292))0.359(0.040)0.374(0.040)
UPG-Gaussian268.63(263.40)−531.26(−524.71)0.885(0.003)(0.704(0.011))0.999(0.051)18.73(0.246)
UL-Clayton205.11−404.222.781(0.229)2.598(0.431)0.550(0.051)
UL-Gumbel291.90−577.801.741(0.080)1.000(0.001)2.599(0.699)
UL-Frank238.44−470.885.124(0.314)1.001(0.051)2.721(0.085)
UL-Gaussian266.37−526.730.789(0.012)2.718(0.005)1.001(0.001)

BAI & BFPUPG-Clayton205.35( 110.6)−404.71(−219.12)0.003(0.001)(1.953(0.080))0.380(0.001)0.247(0.001)
UPG-Gumbel285.08(284.30)−564.16(−566.52)1.886(0.076)( 1.968(0.072))0.0003(2.15)1.026(2.869)
UPG-Frank268.82(234.00)−531.64(−465.93)4.064(0.362)(5.445(0.342))0.519(0.001)0.188(0.001)
UPG-Gaussian277.35(268.40)−548.70(−534.71)0.889(0.007)(0.708(0.017))0.992(0.040)18.50(0.016)
UL-Clayton180.81−355.622.564(0.196)2.697(0.023)0.550(0.001)
UL-Gumbel285.01−564.031.902(0.078)7.616(0.909)2.718(0.001)
UL-Frank201.85−397.694.244(0.292)1.001(0.102)2.721(0.081)
UL-Gaussian270.89−535.780.793(0.012)2.717(0.036)1.001(0.002)

BAI & WCUPG-Clayton369.83(333.20)−733.67(−664.36)0.042(0.001)(3.066(0.183))0.895(0.001)0.033(0.002)
UPG-Gumbel410.60(348.00)−815.20(−693.97)1.600(0.046)(2.212(0.082))0.005(0.001)1.204(0.001)
UPG-Frank386.81(370.60 )−767.61(−739.24)3.743(0.279)(7.660(0.420))0.014(0.001)1.106(0.001)
UPG-Gaussian425.21(411.20)−844.42(−820.33)0.688(0.097)(0.809(0.013))0.431(0.392)0.870(0.554)
UL-Clayton379.13−752.263.641(0.276)1.909(0.081)0.690(0.002)
UL-Gumbel379.71−753.411.945(0.062)1.098(0.351)2.718(0.002)
UL-Frank387.54−769.088.249(0.628)1.145(0.203)2.719(0.016)
UL-Gaussian423.92−841.840.819(0.038)1.171(0.297)4.757(0.231)

BFP & WCUPG-Clayton55.14( 17.65)−104.27(−33.29)0.003(0.001)(0.670(0.049))0.784(0.002)0.058(0.003)
UPG-Gumbel64.59(64.15)−123.18(−126.30)1.266(0.043)( 1.293(0.045))0.001(1.077)19.51(1.459)
UPG-Frank64.52(46.38 )−123.04(−90.76)1.332(0.342)( 2.062(0.275))0.765(0.008)0.070(0.005)
UPG-Gaussian68.53(66.07)−131.06(−130.13)0.780(0.018)(0.399(0.037))0.981(0.180)97.50(0.002)
UL-Clayton33.10−60.201.014(0.139)2.719(0.001)2.718(0.002)
UL-Gumbel64.58−123.151.266(0.044)11.717(0.594)2.710(0.159)
UL-Frank35.51−65.022.183(0.419)2.747(0.260)2.749(0.292)
UL-Gaussian64.92−123.850.503(0.030)2.718(0.007)1.000(0.001)

Table 4

Cramer-von Mises statistics (p-value) for the chosen copula models

ClaytonGumbelFrankGaussian
BMI & BAI0.55(0.025)0.09(0.609)0.03(0.998)0.04(0.986)
BMI & BFP0.77(0.006)0.15(0.335)0.24(0.171)0.14(0.401)
BMI & WC1.07(0.001)0.15(0.321)0.11(0.531)0.19(0.253)
BAI & BFP0.70(0.005)0.03(0.999)0.12(0.876)0.07(0.890)
BAI & WC0.43(0.071)0.34(0.080)0.14(0.413)0.11(0.580)
BFP & WC0.48(0.019)0.60(0.006)0.71(0.020)0.69(0.013)

UPG-ClaytonUPG-GumbelUPG-FrankUPG-Gaussian

BMI & BAI0.45(0.542)0.02(0.941)0.04(0.929)0.03(0.932)
BMI & BFP0.51(0.494)0.14(0.765)0.16(0.758)0.11(0.829)
BMI & WC0.73(0.257)0.09(0.827)0.13(0.768)0.18(0.720)
BAI & BFP0.35(0.685)0.03(0.923)0.07(0.821)0.04(0.896)
BAI & WC0.50(0.480)0.20(0.708)0.23(0.699)0.08(0.843)
BFP & WC0.38(0.713)0.28(0.741)0.29(0.755)0.24(0.782)

UL-ClaytonUL-GumbelUL-FrankUL-Gaussian

BMI & BAI0.37(0.578)0.02(0.916)0.15(0.799)0.04(0.927)
BMI & BFP0.85(0.168)0.14(0.754)0.42(0.551)0.13(0.804)
BMI & WC0.79(0.276)0.11(0.824)0.41(0.537)0.18(0.707)
BAI & BFP0.54(0.508)0.03(0.912)0.34(0.606)0.06(0.870)
BAI & WC0.34(0.564)0.20(0.690)0.82(0.193)0.09(0.830)
BFP & WC0.68(0.440)0.28(0.730)0.62(0.451)0.26(0.762)

Table 5

Summary of the best base or distorted copula model with its Kendall’s tau value and its coefficient of upper tail dependence

Good distorted (or base) modelτ̂T(τ̂)λ̂T,U(λ̂U)
BMI & BAIUPG-Gumbel(Frank)0.510(0.292)0.722(−)
BMI & BFPUPG-Gaussian(Gaussian)0.516(0.466)−(−)
BMI & WCUPG-Gumbel(Gumbel)0.527(0.505)0.454(0.590)
BAI & BFPUPG-Gumbel(Gumbel)0.644(0.492)0.556(0.578)
BAI & WCUPG-Gaussian(Gaussian)0.586(0.510)−(−)
BFP & WCUPG-Gaussian(None)0.220(−)−(−)

References
  1. Ahuja JC and Nash SW (1967). The generalized Gompertz-Verhulst family of distributions. The Indian Journal of Statistics, Series A, 29, 141-156.
  2. Aldhufairi FA, Samanthi RG, and Sepanski JH (2020). New families of bivariate copulas via unit lomax distortion. Risks, 8, 1-19.
    CrossRef
  3. Aldhufairi FA and Sepanski JH (2020). New families of bivariate copulas via unit weibull distortion. Journal of Statistical Distributions and Applications, 7, 1-20.
    CrossRef
  4. Charpentier A (2008). Dynamic dependence ordering for Archimedean copulas and distorted copulas. Kybernetika, 44, 777-794.
  5. Cherubini U, Luciano E, and Vecchiato W (2004). Copula Methods in Finance, Hoboken, NJ, John Wiley & Sons.
    CrossRef
  6. Di Bernardino E and Rulliere D (2013). Distortions of multivariate distribution functions and associated level curves, applications in multivariate risk theory. Insurance: Mathematics and Economics, 53, 190-205.
  7. Durante F and Sempi C (Array). Copula and semicopula transforms. International Journal of Mathematics and Mathematical Sciences, 645-655.
    CrossRef
  8. Durante F, Foschi R, and Sarkoci P (2010). Distorted copulas: Constructions and tail dependence. Communications in Statistics - Theory and Methods, 39, 2288-2301.
    CrossRef
  9. Frees EW and Valdez EA (1998). Understanding relationships using copulas. North American Actuarial Journal, 2, 1-25.
    CrossRef
  10. Gompertz B (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513-583.
    CrossRef
  11. Genest C, Remillard B, and Beaudoin D (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 199-213.
  12. Gudendorf G and Segers J (2010). Extreme-value copulas. Copula Theory and Its Applications, (pp. 127-145), Springer.
    CrossRef
  13. Hua L and Joe H (2013). Intermediate tail dependence: A review and some new results. Stochastic Orders in Reliability and Risk Lecture Notes in Statistics, 208, 291-311.
    CrossRef
  14. Jaworski P, Durante F, Hardle WK, and Rychlik T (2010). Copula Theory and Its Applications, New York, Springer.
    CrossRef
  15. Joe H (2015). Dependence Modeling with Copulas, Boca Raton, FL, CRC Press.
  16. Joe H (1997). Multivariate Models and Multivariate Dependence Concepts, London, Chapman & Hall.
  17. McNeil AJ, Frey R, and Embrechts P (2006). Quantitative risk management: Concepts, techniques, and tools. Journal of the American Statistical Association, 101, 1731-1732.
    CrossRef
  18. Morillas PM (2005). A method to obtain new copulas from a given one. Metrika, 61, 169-184.
    CrossRef
  19. Nelsen RB (2006). An Introduction to Copulas, New York, Springer.
  20. Saporu FWO and Gongsin IE (2020). Modeling dependence relationships of anthropometric variables using copula approach. American Journal of Theoretical and Applied Statistics, 9, 245-255.
    CrossRef
  21. Sepanski JH (2020). A note on distortion effects on the strength of bivariate copula tail dependence. Statistics Probability Letters, 166, 108-894.
    CrossRef
  22. Sklar M (1959). Fonctions de repartition a n dimensions et leurs marges. Publications de l섾nstitut Statistique de l셐niversite de Paris, 8, 229-231.
  23. Klugman SA and Parsa R (1999). Fitting bivariate loss distributions with copulas. Insurance: Mathematics and Economics, 24, 139-148.
  24. Valdez EA and Xiao Y (2011). On the distortion of a copula and its margins. Scandinavian Actuarial Journal, 4, 292-317.
    CrossRef
  25. Xie J, Yang J, and Zhu W (2019). A family of transformed copulas with a singular component. Fuzzy Sets and Systems, 354, 20-47.
    CrossRef