Copulas are multivariate distribution functions characterized by uniform marginal distributions. In 1959, Sklar [1] demonstrated that for any multivariate distribution, a copula exists that allows the joint distribution to be expressed as the copula applied to the marginal distributions. Because copulas are independent of the marginal distributions, they provide a convenient framework for addressing various modeling challenges. Copulas have been used in many fields, including finance, engineering, economics, and environmental science [2] - [5] .

Fuzzy implications play a critical role in fuzzy logic, enabling logical reasoning in uncertain situations [6] . Among them, residual implications stand out as a key type and are derived from aggregation functions [7] - [10] . In recent years, many researchers have explored fuzzy implications formed by various aggregation functions [11] - [13] , while others have focused on characterizing residual implications based on different aggregation functions [14] [15] . Additionally, some studies constructed different fuzzy implications from copulas, including probabilistic implications and probability S-implications [16] - [18] .

There is a one-to-one correspondence between left-continuous semicopulas and their derived residual implications [9] . In 1999, a characterization for residual implications derived from left-continuous semicopulas was provided by Demirli et al. [9] . In 2007, Durante et al. [19] extended this work by characterizing residual implications derived from quasi-copulas. More recently, in 2022, Ji et al. [20] further advanced this area by characterizing residual implications derived from copulas through their level curves.

This article aims to study a class of copulas derived from residual implications, where the extensions of the level curves intersect at a point. Specifically, this class represents a comprehensive and asymmetric extension of the Archimedean copula family (4.2.7) as described in [23] , providing greater flexibility for modeling real-world phenomena. The proposed copulas include two dependence parameters, $a$ and $b$, and are able to capture correlation coefficients in the range $\left[-1,1\right]$, i.e. Kendall’s tau and Spearman’s rho. It is well known that multiparameter copulas are capable of modeling a variety of dependency structures, including non-linear, asymmetric and tail dependent relationships that one-parameter copulas cannot adequately capture. In summary, the new model allows for both positive and negative dependencies without imposing constraints on the correlation structure. These important features motivate a theoretical and practical analysis of the proposed copulas.

The remainder of the paper is structured as follows. Section 2 provides a review of the fundamental concepts and key properties related to copulas and residual implications. In section 3, we introduce radial copulas and examine some dependence concepts. Section 4 focuses on estimating the two dependence parameters of radial copulas and includes a simulation study to assess the performance of these estimators. In section 5, we perform an analysis of two real datasets to examine the applications of the proposed radial copulas. Finally, section 6 presents our conclusions.

Copulas are multivariate functions characterized by uniform marginal distributions. Let us first review some of their definitions and properties.

Definition 1 [21] A binary operation $C:{\left[0,1\right]}^2\to \left[0,1\right]$ is defined as a semi-copula if it fulfills the subsequent conditions: [label=(C)]

1. $C\left(0,0\right)=0$ and $C\left(1,1\right)=1$;

2. For every $u,v\in \left[0,1\right]$, it holds that $C\left(u,1\right)=u$ and $C\left(1,v\right)=v$;

3. $C$ is nondecreasing in both variables.

Definition 2 [22] A binary operation $C:{\left[0,1\right]}^2\to \left[0,1\right]$ is defined as a quasi-copula if it is a semicopula and also adheres to the 1-Lipschitz condition. Specifically, this implies that the following inequality is valid for all $u_1,u_2,v_1,v_2\in \left[0,1\right]$: $\left|C\left(u_1,v_1\right)-C\left(u_2,v_2\right)\right|\le \left|u_1-u_2\right|+\left|v_1-v_2\right|$.

Definition 3 [23] A binary operation $C:{\left[0,1\right]}^2\to \left[0,1\right]$ is defined as a copula if it is a semicopula and also fulfills the condition of being 2-increasing. Specifically, this indicates that for any $u_1,u_2,v_1,v_2\in \left[0,1\right]$ where $u_1\le u_2$ and $v_1\le v_2$, the following inequality is valid: $C\left(u_2,v_2\right)-C\left(u_2,v_1\right)-C\left(u_1,v_2\right)+C\left(u_1,v_1\right)\ge 0$.

Residual implications derived from various classes of left-continuous semicopulas have been examined in [9] [19] . For any left-continuous semicopula $C$, the derived residual implication, denoted as $R_C:{\left[0,1\right]}^2\to \left[0,1\right]$, is defined by

$R_C\left(u,w\right)=\text{sup}\left\{v\in \left[0,1\right]|C\left(u,v\right)\le w\right\}.$ (1)

This function $R_C$ is referred to as the residuum of $C$ or the $C$-residuum [19] .

Proposition 4 [9] If $C:{\left[0,1\right]}^2\to \left[0,1\right]$ is a left-continuous semicopula, then the corresponding $C$ residuum $R_C$ has the properties listed below:

1. $R_C\left(u,w\right)=1$ if and only if $u\le w$;

2. $R_C\left(1,w\right)=w$ for all $w\in \left[0,1\right]$;

3. If $u_1,u_2,w\in \left[0,1\right]$ with $u_1\le u_2$, then it holds that $R_C\left(u_1,w\right)\ge R_C\left(u_2,w\right)$;

4. If $u,w_1,w_2\in \left[0,1\right]$ with $w_1\le w_2$, then it holds that $R_C\left(u,w_1\right)\le R_C\left(u,w_2\right)$;

5. The first variable of the $R_C$ function is left-continuous;

6. The second variable of the $R_C$ function is right continuous.

On the contrary, if a function $R:{\left[0,1\right]}^2\to \left[0,1\right]$ fulfills conditions (R1) through (R6), then there exists a left-continuous semicopula $C_R$ where $R=R_{C_R}$. Additionally, the definition of $C_R$ is as follows:

$C_R\left(u,v\right)=\text{inf}\left\{w\in \left[0,1\right]|R\left(u,w\right)\ge v\right\}.$ (2)

The set of all functions $R:{\left[0,1\right]}^2\to \left[0,1\right]$ that fulfill conditions (R1) through (R6) is represented by $ℛ$. Given any $R\in ℛ$, the associated deresiduum, referred to as the $R$-deresiduum, is denoted by $C_R$ [19] . For simplicity, when the context is clear, $R_C$ and $C_R$ are often abbreviated as $R$ and $C$, respectively.

Similar to the t-norms, the adjointness condition of the

$C\left(u,v\right)\le w\iff R\left(u,w\right)\ge v$ (3)

applies to any left-continuous semicopula $C$ and its residuum $R$ [9] .

Proposition 5 [19] [20] Let $C:{\left[0,1\right]}^2\to \left[0,1\right]$ be a quasi-copula. The $C$-residuum $R_C$ fulfills conditions (R1) through (R6) and two additional properties:

1. For all $u,w,a\in \left[0,1\right]$ such that $u+a\le 1$ and $w+a\le 1$, it holds that $R\left(u+a,w+a\right)\ge R\left(u,w\right)$;

2. For all $u,w,a\in \left[0,1\right]$ with the condition that $u\ge w+a$, it holds that $R\left(u,w+a\right)\ge R\left(u,w\right)+a$.

By contrast, if $R:{\left[0,1\right]}^2\to \left[0,1\right]$ fulfills conditions (R1) through (R8), then the $R$-deresiduum $C_R$ is a quasi-copula.

The level set of $R\in ℛ$ for a constant $t\in \left[0,1\right]$ is defined as $\left\{\left(u,w\right)|R\left(u,w\right)=t\right\}$. Let $R:{\left[0,1\right]}^2\to \left[0,1\right]$ be a function that satisfies conditions (R1) through (R8). For $0\le t<1$, if $R\left(u,w_1\right)=R\left(u,w_2\right)=t$, then according to condition (R8), we conclude that $w_1=w_2$. Consequently, the level set of $R$ associated with $t\in \left[0,1\right)$ is made up of points lying on a curve, known as the level curve of $R$ [20] . For $t=1$, the level set of $R$ is defined as $\left\{\left(u,w\right)|u\le w\le 1,0\le u\le 1\right\}$. The boundary curve of this level set, $\left\{\left(u,w\right)|w=u,0\le u\le 1\right\}$, is referred to as the level curve of $R$ for the constant 1. We often denote the level curve of $R$ as $w=L_t\left(u\right)$, where $0\le t\le 1$.

Proposition 6 [20] Let $C:{\left[0,1\right]}^2\to \left[0,1\right]$ be a quasi-copula, and let $R$ represent its residuum. Suppose that $w=L_t\left(u\right)$ represents a level curve of $R$ for $t\in \left[0,1\right]$. For $u_1,u_2\in Dom\left(L_t\left(u\right)\right)$ such that $u_1\le u_2$, we have $0\le L_t\left(u_2\right)-L_t\left(u_1\right)\le u_2-u_1$.

The following proposition gives a necessary and sufficient condition for the $R$-deresiduum $C$ to be a copula.

Proposition 7 [20] Let $R:{\left[0,1\right]}^2\to \left[0,1\right]$ satisfy conditions (R1) through (R8). Let $\left\{w=L_t\left(u\right)|t\in \left[0,1\right]\right\}$ denote the set of level curves of $R$. The $R$-deresiduum $C$ is a copula if and only if each level curve of $R$ has a domain that is an interval, and the inequality $L_{v_2}\left(u_1\right)-L_{v_1}\left(u_1\right)\le L_{v_2}\left(u_2\right)-L_{v_1}\left(u_2\right)$ holds for $u_1,u_2,v_1,v_2\in \left[0,1\right]$, where $u_1\le u_2$, $v_1\le v_2$, and $L_{v_j}\left(u_i\right)$ exists for all $i,j\in \left\{1,2\right\}$.

The domain of any level curve is either $\left(a,1\right]$ or $\left[a,1\right]$, where $0\le a\le 1$, for the residual implication derived from a copula. Notably, the vertical distance between two level curves is consistently greater on the right side than on the left [20] .

This section introduces a class of copulas derived from residual implications, which are characterized by extensions of level curves that intersect at a point. This class includes two dependence parameters, providing greater flexibility in modeling dependencies.

For the product copula $\text{Π}\left(u,v\right)=uv$, the corresponding residual implication is given by $R_{\text{Π}}\left(u,w\right)=\{\begin{array}{ll}\frac{w}{u},\hfill & \text{if}u>w\hfill \\ 1,\hfill & \text{if}u\le w\hfill \end{array}$. For $0\le v\le 1$, the level curves of $R_{\text{Π}}$ can

be expressed by the equation $w=L_v\left(u\right)=vu$, where $0<u\le 1$. Clearly, for a given value of $v$, the level curves of $R_{\text{Π}}$ are segments of lines with a slope of $v$, all intersecting at the origin.

According to Proposition 6, if the level curves lie on a line, then the slope of the level curves of residual implications lies within the interval $\left[0,1\right]$. Consequently, when the level curves of a residual implication derived from a copula are straight lines intersecting at a common point, this intersection point is represented as $\left(-a,-b\right)$, where $0\le b\le a$. Given the intersection point $\left(-a,-b\right)$ of the extended level curves and noting that they always pass through $\left(1,v\right)$, we derive the

equation of the line on which these level curves lie as $\frac{w+b}{u+a}=\frac{v+b}{1+a}$. This simplifies to $w=\frac{uv+bu+av-b}{1+a}$. Thus, $R_{a,b}\left(u,w\right)=\{\begin{array}{ll}\frac{aw+w-bu+b}{u+a},\hfill & \text{if}u>w\hfill \\ 1,\hfill & \text{if}u\le w\hfill \end{array}$. It follows from Propositions 5 and 7 that the function $R_{a,b}$ is a residual implication derived from a copula.

For $a,b$ in $\bar{R}$ such that $0\le b\le a$, let $R_{a,b}:{\left[0,1\right]}^2\to \left[0,1\right]$ be a function given by

$R_{a,b}\left(u,w\right)=\{\begin{array}{ll}\frac{aw+w-bu+b}{u+a},\hfill & \text{if}u>w,\hfill \\ 1,\hfill & \text{if}u\le w.\hfill \end{array}$ (4)

Then the corresponding copula $C_{a,b}:{\left[0,1\right]}^2\to \left[0,1\right]$ is given by

$C_{a,b}\left(u,v\right)=\{\begin{array}{ll}0,\hfill & \text{if}v<\frac{b\left(1-u\right)}{u+a},\hfill \\ \frac{uv+bu+av-b}{1+a},\hfill & \text{if}\frac{b\left(1-u\right)}{u+a}\le v<\frac{\left(1+a-b\right)u+b}{u+a},\hfill \\ u,\hfill & \text{if}v\ge \frac{\left(1+a-b\right)u+b}{u+a}.\hfill \end{array}$ (5)

We refer to the copula $C_{a,b}$ as a radial copula. The left side of Figure 1 shows some level curves of $R_{a,b}$, while the right side displays the graph of the radial copula $C_{a,b}$.

Remark 1 Here are some special cases of the radial copula: [label = ()]

1. If $a=0$, then $b=0$, i.e., $C_{0,0}\left(u,v\right)=\text{Π}\left(u,v\right)=uv$.

2. If $a=b<+\infty$, then $C_{a,a}\left(u,v\right)=\text{max}\left(\left(uv+au+av-a\right)/\left(1+a\right),0\right)$, which is equal to the Archimedean copula family (4.2.7) in [23] .

3. If $a=+\infty$ and $0\le b<+\infty$, then $\underset{a\to +\infty }{\text{lim}}{C}_{a,b}\left(u,v\right)=M\left(u,v\right)$.

4. If $k\in \left[0,1\right]$ and $b=ka$, then

$\underset{a\to +\infty }{\text{lim}}{C}_{a,ka}\left(u,v\right)=\{\begin{array}{ll}0,\hfill & \text{if}v<k\left(1-u\right),\hfill \\ ku+v-k,\hfill & \text{if}k\left(1-u\right)\le v<\left(1-k\right)u+k,\hfill \\ u,\hfill & \text{if}v\ge \left(1-k\right)u+k.\hfill \end{array}$

The latter part of the formula is equal to the copula $C_{\theta }$ studied in [20, Example 5.14]. In particular, if $k=0$, then $\underset{a\to +\infty }{\text{lim}}{C}_{a,0}\left(u,v\right)=M\left(u,v\right)$. If $k=1$, then $\underset{a\to +\infty }{\text{lim}}{C}_{a,a}\left(u,v\right)=W\left(u,v\right)$.

The class of radial copulas is obviously an asymmetric extension of the Archimedean copula family (4.2.7), as presented in [23] . Notably, the Archimedean copula family (4.2.7) does not include the Frechet-Hoeffding upper bound $M$. As a result, the class of radial copulas provides a comprehensive extension of the Archimedean copula family (4.2.7). In Figure 2 and Figure 3 , we show the radial copula plots and their contour plots for four different parameter value sets $a$ and $b$.

There are various methods employed to generate observations $\left(x,y\right)$ for a pair of random variables $\left(X,Y\right)$ characterized by a joint distribution function $H$. One such method involves utilizing the copula function and the concept of conditional probability, referred to as the conditional distribution method [23] . This approach proposes an algorithm for generating random samples from the copula $C_{a,b}$ using the conditional copula $V$ given $U=u$. The algorithm is as follows:

1. Generate two independent uniform random variables $u_i$ and $t_i$ from the interval $\left[0,1\right]$.

2. Define $v_i=c_{u_i}^{\left(-1\right)}\left(t_i\right)$, where $c_{u_i}^{\left(-1\right)}$ represents a quasi-inverse of the function $c_{u_i}$, and also $c_{u_i}\left(v_i\right)=\frac{\partial C_{a,b}\left(u_i,v_i\right)}{\partial u_i}$. Specifically, the conditional distribution function $c_{u_i}\left(v_i\right)$ is given by

$c_{u_i}\left(v_i\right)=\frac{\partial C_{a,b}\left(u_i,v_i\right)}{\partial u_i}=\{\begin{array}{ll}0,\hfill & \text{if}{v}_i<\frac{b\left(1-u_i\right)}{u_i+a},\hfill \\ \frac{v_i+b}{1+a},\hfill & \text{if}\frac{b\left(1-u_i\right)}{u_i+a}\le v_i<\frac{\left(1+a-b\right)u_i+b}{u_i+a},\hfill \\ 1,\hfill & \text{if}{v}_i\ge \frac{\left(1+a-b\right)u_i+b}{u_i+a},\hfill \end{array}$

and its quasi-inverse $c_{u_i}^{\left(-1\right)}\left(t_i\right)$ is given by

$c_{u_i}^{\left(-1\right)}\left(t_i\right)=\{\begin{array}{ll}\frac{b\left(1-u_i\right)}{u_i+a},\hfill & \text{if}{t}_i<\frac{b}{u_i+a},\hfill \\ \left(1+a\right)t_i-b,\hfill & \text{if}\frac{b}{u_i+a}\le t_i<\frac{u_i+b}{u_i+a},\hfill \\ \frac{\left(1+a-b\right)u_i+b}{u_i+a},\hfill & \text{if}{t}_i\ge \frac{u_i+b}{u_i+a}.\hfill \end{array}$

3. Repeat steps i and ii $n$ times to generate a set of independent and identically distributed pairs $\left(u_i,v_i\right)$ from the copula $C_{a,b}$.

Figure 4 displays scatter plots derived from 500 simulated data points, which were generated using the previously described algorithm with four distinct sets of parameter values for $a$ and $b$. Figures 4(c) and 4(d) correspond to the scatter plots of copulas in [25] with $\theta =0.9$ (right) and $\theta =0.4$ (left), respectively.

For any copula $C$, we have the decomposition

$C\left(u,v\right)=A_C\left(u,v\right)+S_C\left(u,v\right),$

where

$A_C\left(u,v\right)={\displaystyle {\int }_0^u{\displaystyle {\int }_0^v\frac{{\partial }^2}{\partial s\partial t}C\left(s,t\right)\text{d}t\text{d}s}}\text{and}{S}_C\left(u,v\right)=C\left(u,v\right)-A_C\left(u,v\right),$ (6)

with $A_C$ representing the absolutely continuous component and $S_C$ the singular component of the copula $C$.

If a copula $C$ satisfies $C=A_C$ on ${\left[0,1\right]}^2$, that is, if $C$ can be represented as a joint distribution with density $\frac{{\partial }^{2}C\left(u,v\right)}{\partial u\partial v}$, then $C$ is absolutely continuous. If $C=S_C$ on ${\left[0,1\right]}^2$, meaning that $\frac{{\partial }^{2}C\left(u,v\right)}{\partial u\partial v}=0$ almost everywhere, then $C$ is classified as singular [23] . Conversely, if this condition does not hold, $C$ can be decomposed into an absolutely continuous component $A_C$ and a singular component $S_C$.

Proposition 8 The radial copula contains an absolutely continuous $A_{C_{a,b}}$ and singular components $S_{C_{a,b}}$. Thus we have

$A_C\left(u,v\right)=\{\begin{array}{ll}\frac{uv}{1+a},\hfill & \text{if}\frac{b\left(1-u\right)}{u+a}\le v<\frac{\left(1+a-b\right)u+b}{u+a},\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}$

and consequently

$S_C\left(u,v\right)=\{\begin{array}{ll}0,\hfill & \text{if}v<\frac{b\left(1-u\right)}{u+a},\hfill \\ \frac{bu+av-b}{1+a},\hfill & \text{if}\frac{b\left(1-u\right)}{u+a}\le v<\frac{\left(1+a-b\right)u+b}{u+a},\hfill \\ u,\hfill & \text{if}v\ge \frac{\left(1+a-b\right)u+b}{u+a}.\hfill \end{array}$

Proof. The proof is immediate, so it is omitted.

Thus, the $C_{a,b}$-measure of the singular component of $C_{a,b}$ is expressed as $S_{C_{a,b}}\left(1,1\right)=1$. In other words, if $U$ and $V$ are uniformly distributed random variables across the interval [0, 1], and their joint distribution is characterized by the copula $C_{a,b}$, then it follows that $P\left[U=V\right]=1$.

In the following proposition, we explore the statistical properties of the copula $C_{a,b}$. Specifically, we examine the behavior of classical concordance measures, including Kendall’s $\tau$ and Spearman’s $\rho$, under the copula $C_{a,b}$.

Let $X$ and $Y$ be continuous random variables associated with the copula $C$, where $F$ and $G$ represent their respective marginal distribution functions. Using the probability transformations $u=F\left(x\right)$ and $v=G\left(y\right)$, Kendall’s $\tau$ for $X$ and $Y$ can be expressed as [23] :

${\tau }_{X,Y}=1-4{\displaystyle {\iint }_{I^2}\frac{\partial }{\partial u}C\left(u,v\right)\frac{\partial }{\partial v}C\left(u,v\right)\text{d}u\text{d}v},$ (7)

and Spearman’s $\rho$ as [23] :

${\rho }_{X,Y}=12{\displaystyle {\iint }_{I^2}C\left(u,v\right)\text{d}u\text{d}v}-3.$ (8)

Proposition 9 Let $X$ and $Y$ be continuous random variables of copula $C_{a,b}$, where $0\le b\le a$. Then

1. Kendall’s $\tau$ for the random variables $X$ and $Y$ is given by

${\tau }_{a,b}=\left(2a-4b\right)\left(1-a\ln \frac{1+a}{a}\right),$ (9)

which takes values between −1 and 1.

2. Spearman’s $\rho$ for the random variables $X$ and $Y$ is given by

${\rho }_{a,b}=3\left(2a^2+a-2b-4ab\right)+6\left(2a^{2}b+2ab-a^2-a^3\right)\ln \frac{1+a}{a},$ (10)

which also takes values between −1 and 1.

Proof.

1. Given that the copula $C_{a,b}$ contains both an absolutely continuous and a singular component, we apply formula 7 to compute Kendall’s ${\tau }_{a,b}$. Then

$\frac{\partial }{\partial u}{C}_{a,b}\left(u,v\right)\frac{\partial }{\partial v}{C}_{a,b}\left(u,v\right)=\{\begin{array}{ll}\frac{\left(u+a\right)\left(v+b\right)}{{\left(1+a\right)}^2},\hfill & \text{if}\frac{b\left(1-u\right)}{u+a}\le v<\frac{\left(1+a-b\right)u+b}{u+a},\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}$

and hence

$\begin{array}{c}{\tau }_{a,b}=1-4{\displaystyle {\iint }_{I^2}\frac{\partial }{\partial u}C\left(u,v\right)\frac{\partial }{\partial v}C\left(u,v\right)\text{d}u\text{d}v}\\ =1-4{\displaystyle {\int }_0^1\left({\displaystyle {\int }_{\frac{b\left(1-u\right)}{u+a}}^{\frac{\left(1+a-b\right)u+b}{u+a}}\frac{\left(u+a\right)\left(v+b\right)}{{\left(1+a\right)}^2}\text{d}v}\right)\text{d}u}\\ =1-4\left(\frac{1}{2\left(1+a\right)}{\displaystyle {\int }_0^1\frac{\left(1+a-2b\right)u^2+2bu}{u+a}\text{d}u}+\frac{b}{1+a}{\displaystyle {\int }_0^{1}u\text{d}u}\right)\\ =1-4\left(\frac{1}{4}-\frac{1}{2}a+\frac{1}{2}{a}^2\ln \frac{1+a}{a}+b\left(1-a\ln \frac{1+a}{a}\right)\right)\\ =\left(2a-4b\right)\left(1-a\ln \frac{1+a}{a}\right).\end{array}$

In particular, if $a=+\infty$ and $b=0$, then we have

$\underset{a\to +\infty }{\lim }{\tau }_{a,0}=2a\left(1-a\ln \frac{1+a}{a}\right),$

which can be written as

$\begin{array}{c}\underset{t\to 0}{\lim }{\tau }_{\frac{1}{t},0}=\frac{2t-2\ln \left(1+t\right)}{t^2}\\ =\frac{2t-2\left(t-\frac{1}{2}{t}^2+O\left(t^3\right)\right)}{t^2}\\ =1.\end{array}$

Similarly, if $a=b=+\infty$, then we obtain $\underset{a\to +\infty }{\lim }{\tau }_{a,a}=-2a\left(1-a\ln \frac{1+a}{a}\right)=-1$.

2. We use formula 8 and the copula $C_{a,b}$ to derive Spearman’s ${\rho }_{a,b}$. Then we have

$\begin{array}{c}{\rho }_{a,b}=12{\displaystyle {\iint }_{I^2}C\left(u,v\right)\text{d}u\text{d}v}-3\\ =12\left({\displaystyle {\int }_0^1\left({\displaystyle {\int }_{\frac{b\left(1-u\right)}{u+a}}^{\frac{\left(1+a-b\right)u+b}{u+a}}\frac{uv+bu+av-b}{1+a}\text{d}v}+{\displaystyle {\int }_{\frac{\left(1+a-b\right)u+b}{u+a}}^{1}u\text{d}v}\right)\text{d}u}\right)-3\\ =12\left({\displaystyle {\int }_0^1\left(\frac{\left(1-a+2b\right)u^2}{2\left(u+a\right)}+\frac{\left(a-b\right)u}{u+a}\right)\text{d}u}\right)-3\\ =12\left({\displaystyle {\int }_0^1\left(\frac{1-a+2b}{2}\left(u-a\right)+\frac{a^2}{2\left(u+a\right)}+a-b+\frac{a}{u+a}\right)\text{d}u}\right)-3\\ =12\left(\frac{2a^2+a-2b-4ab+1}{4}+\frac{1}{2}\left(2a^{2}b+2ab-a^2-a^3\right)\ln \frac{1+a}{a}\right)-3\end{array}$

$=3\left(2a^2+a-2b-4ab\right)+6\left(2a^{2}b+2ab-a^2-a^3\right)\ln \frac{1+a}{a}.$

In particular, if $a=+\infty$ and $b=0$, then we have

$\underset{a\to +\infty }{\text{lim}}{\rho }_{a,0}=3\left(2a^2+a\right)+6\left(-a^2-a^3\right)\text{ln}\frac{1+a}{a},$

which can be written as

$\begin{array}{c}\underset{t\to 0}{\lim }{\rho }_{\frac{1}{t},0}=3\left(\frac{2}{t^2}+\frac{1}{t}\right)+6\left(-\frac{1}{t^2}-\frac{1}{t^3}\right)\ln \left(1+t\right)\\ =3\left(\frac{2}{t^2}+\frac{1}{t}\right)+6\left(-\frac{1}{t^2}-\frac{1}{t^3}\right)\left(t-\frac{1}{2}{t}^2+\frac{1}{3}{t}^3+O\left(t^4\right)\right)\\ =\frac{t^2-2t^3}{t^2}\\ =1.\end{array}$

Similarly, if $a=b=+\infty$, then we get

$\underset{a\to +\infty }{\text{lim}}{\rho }_{a,a}=3\left(-2a^2-a\right)+6\left(a^2+a^3\right)\text{ln}\frac{1+a}{a}=-1$.

The full $\left[-1,1\right]$ range for Kendall’s ${\tau }_{a,b}$ and Spearman’s ${\rho }_{a,b}$ is theoretically proven in Proposition 9, which establishes that both ${\tau }_{a,b}$ and ${\rho }_{a,b}$ are constrained within this range for all radial copulas. In practice, the $a$ parameter controls the overall strength of the dependence between the variables, with larger values indicating stronger dependence. The $b$ parameter, on the other hand, primarily affects the tail dependence, with larger values leading to greater correlation at the extreme values. Together, $a$ and $b$ provide flexibility in modeling both general dependence and extreme comovements. In Figure 5 , we plot Spearman’s $\rho$ and Kendall’s $\tau$ as functions of the dependency parameters $a$ and $b$, where $b$ is constrained by the condition $0\le b\le a$. Specifically, it is set as $b=\text{min}\left(a-10,a\right)$ and further adjusted to 0 if $b<0$.

In this section, we estimate the two dependence parameters $\left(a,b\right)$ of the copula $C_{a,b}$ using the $\left(\tau ,\rho \right)$-inversion method and bivariate L-moments. Furthermore, a simulation study is performed to measure the performance of these two estimation methods.

Tsukahara [24] introduced the rank approximate Z-estimation (RAZ) estimators, which are derived from Kendall’s tau ( $\tau$) and Spearman’s rho ( $\rho$). These estimators are referred to as the $\tau$-score and $\rho$-score RAZ estimators. Let us assume where $r=1$, with ${\hat{\tau }}_n$ and ${\hat{\rho }}_n$ denoting the sample estimators of Kendall’s tau ( $\tau$) and Spearman’s rho ( $\rho$), respectively. The $\tau$-inverse estimator ${\hat{\theta }}_{\tau }$ and the $\rho$-inverse estimator ${\hat{\theta }}_{\rho }$ of $\theta$ are defined using a method that is analogous to the method of moments:

${\hat{\theta }}_{\tau }={\tau }^{-1}\left({\hat{\tau }}_n\right)\text{and}{\hat{\theta }}_{\rho }={\rho }^{-1}\left({\hat{\rho }}_n\right).$

When $r=2$, the parameters $\theta =\left({\theta }_1,{\theta }_2\right)$ can be estimated by solving the following system of equations:

$\tau \left({\theta }_1,{\theta }_2\right)={\hat{\tau }}_n,$

$\rho \left({\theta }_1,{\theta }_2\right)={\hat{\rho }}_n.$

We refer to the solution obtained from the preceding system as the $\left(\tau ,\rho \right)$-inversion estimator of $\theta$ [25] . This method is employed to estimate the parameters $\left(a,b\right)$ of the radial copula $C_{a,b}$ as shown in the formula 5. The corresponding Kendall’s tau ( $\tau$) and Spearman’s rho ( $\rho$) are computed from the formulas 9 and 10, respectively:

$\tau \left(a,b\right)=\left(2a-4b\right)\left(1-a\ln \frac{1+a}{a}\right),$

$\rho \left(a,b\right)=3\left(2a^2+a-2b-4ab\right)+6\left(2a^{2}b+2ab-a^2-a^3\right)\text{ln}\frac{1+a}{a}.$

The solution of the following system is defined as the $\left(\tau ,\rho \right)$ inversion estimator for the parameters $\left(a,b\right)$:

$\begin{array}{l}\tau \left(a,b\right)={\hat{\tau }}_n,\\ \rho \left(a,b\right)={\hat{\rho }}_n.\end{array}$ (11)

Another key method for estimating bivariate copula models with multiparameter is the application of bivariate L-moments. To begin, we offer a brief overview of univariate L-moments. Hosking [26] proposed L-moments, denoted by ${\lambda }_k$, as an alternative to the traditional central moments ${\mu }_k=\mathbb{E}\left[{\left(Y-\mu \right)}^k\right]$, which are derived from the distribution function (df) $F_Y$ of the random variable $Y$. Using the quantile function $F_Y^{-1}$, the $L$ moment ${\lambda }_k$ can be represented as follows:

${\lambda }_k={\displaystyle {\int }_{\mathbb{I}}{F}_Y^{-1}\left(u\right)P_{k-1}\left(u\right)\text{d}u},$

where $P_k$ denotes the shifted Legendre polynomial [27] . In the following, we will use the first three shifted Legendre polynomials:

$P_0\left(u\right)=1,P_1\left(u\right)=2u-1,P_2\left(u\right)=6u^2-6u+1.$

The bivariate L-moments (BLM) extend univariate L-moments to the multivariate setting. Next we introduce the fundamental notations and definitions associated with BLM. Let $X^{\left(1\right)}$ and $X^{\left(2\right)}$ represent two random variables with finite means, the marginal distributions $F_1$ and $F_2$, as well as the respective sequences of $L$ moments denoted by ${\lambda }_k^{\left(1\right)}$ and ${\lambda }_k^{\left(2\right)}$. Serfling et al. [28] introduced the k-th L-comoment of $X^{\left(1\right)}$ in relation to $X^{\left(2\right)}$, which corresponds to the covariance between $X^{\left(1\right)}$ and $P_k\left(F_2\left(X^{\left(2\right)}\right)\right)$ for values of $k\ge 1$. This relationship can be expressed as follows:

${\lambda }_{k\left[12\right]}=\text{Cov}\left(X^{\left(1\right)},P_k\left(F_2\left(X^{\left(2\right)}\right)\right)\right),$

with asymmetric counterpart ${\lambda }_{k\left[21\right]}$. If $F$ is assumed to be part of a parametric family of distribution functions, the k-th L-comoment ${\lambda }_{k\left[12\right]}$ is influenced by both the parameters of the marginal distributionsd and the dependence structure between $X^{\left(1\right)}$ and $X^{\left(2\right)}$. Given that our primary goal is to estimate the parameters of the copula, it is more practical to apply the k-th L-comoment of $F_1\left(X^{\left(1\right)}\right)$ in relation to $X^{\left(2\right)}$, rather than using ${\lambda }_{k\left[12\right]}$ directly. Thus, we define:

${\delta }_{k\left[12\right]}=\text{Cov}\left(F_1\left(X^{\left(1\right)}\right),P_k\left(F_2\left(X^{\left(2\right)}\right)\right)\right),k=1,3,\cdots .$

This quantity ${\delta }_{k\left[12\right]}$ is referred to as the k-th bivariate copula L-moment of $X^{\left(1\right)}$ in relation to $X^{\left(2\right)}$.

Brahimi et al. [25] introduced the k-th bivariate copula L-moment of $X^{\left(1\right)}$ in relation to $X^{\left(2\right)}$, which can be expressed, for each $k\ge 1$, as

${\delta }_{k\left[12\right]}={\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\left(C\left(u,v\right)-uv\right)\text{d}u\text{d}{P}_k}}\left(v\right).$ (12)

Using this formula, it is possible to set up a system of equations to estimate the parameters of multivariate copula models. For example, if $r=2$ and $C=C_{\theta }$, $\theta =\left({\theta }_1,{\theta }_2\right)$, then based on formula 12, the first two bivariate copula L-moments are provided by [25]

${\delta }_{1\left[12\right]}=2{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1{C}_{\theta }\left(u,v\right)\text{d}u\text{d}v}}-\frac{1}{2},$ (13)

${\delta }_{2\left[12\right]}=6{\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\left(2v-1\right)C_{\theta }\left(u,v\right)\text{d}u\text{d}v}}-\frac{1}{2}.$ (14)

Below is the estimation process for the copula $C_{a,b}$ using the BLM method, structured as follows

1. Step 1: Compute the empirical L-moments. Using the observed sample

${\left(X_i^{\left(1\right)},X_i^{\left(2\right)}\right)}_{i=1,\cdots ,n}$, the empirical L-moment ${\hat{\delta }}_{k\left[12\right]}$ is computed as follows:

${\hat{\delta }}_{k\left[12\right]}=\frac{1}{n}\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{F}_{1:n}^{\text{*}}\left(X_i^{\left(1\right)}\right)P_k\left(F_{2:n}^{\text{*}}\left(X_i^{\left(2\right)}\right)\right),$

where for each $j=1,2$, $F_{j:n}^{\text{*}}$ denotes the rescaled empirical distribution function, given as $F_{j:n}^{\text{*}}=n{F}_{j:n}/\left(n+1\right)$, where $F_{j:n}\left(x_j\right)=\frac{1}{n}{\displaystyle {\sum }_{i=1}^{n}1\left\{X_i^{\left(j\right)}\le x_j\right\}}$ is the empirical distribution function.

2. Step 2: Construct the system of equations. Using formula 12, we can derive a system of $r$ equations as follows:

${\delta }_{k\left[12\right]}\left({\theta }_1,\cdots ,{\theta }_r\right)={\hat{\delta }}_{k\left[12\right]},k=1,\cdots ,r.$

For the radial copula $C_{a,b}$, according to formulas 13 and 14, the two first bivariate copula L-moments are given by:

${\delta }_{1\left[12\right]}\left(a,b\right)=a^2+\frac{a}{2}-2ab-b+\left(2a^{2}b+2ab-a^2-a^3\right)\text{ln}\left(\frac{1+a}{a}\right),$

$\begin{array}{c}{\delta }_{2\left[12\right]}\left(a,b\right)=\frac{a}{2}+6a^2+6a^3-15ab-18a^{2}b+15b^2+18a{b}^2\\ -3\left(1+a\right)\left(a^2+2a^3-2ab-6a^{2}b+2b^2+6a{b}^2\right)\text{ln}\left(\frac{1+a}{a}\right).\end{array}$

3. Step 3: Solve the system

$\begin{array}{l}{a}^2+\frac{a}{2}-2ab-b+\left(2a^{2}b+2ab-a^2-a^3\right)\text{ln}\frac{1+a}{a}={\hat{\delta }}_{1\left[12\right]},\\ \frac{a}{2}+6a^2+6a^3-15ab-18a^{2}b+15b^2+18a{b}^2\\ -3\left(1+a\right)\left(a^2+2a^3-2ab-6a^{2}b+2b^2+6a{b}^2\right)\text{ln}\frac{1+a}{a}={\hat{\delta }}_{2\left[12\right]}.\end{array}$ (15)

The solution $\left(\hat{a},\hat{b}\right)$ obtained is referred to as the BLM estimator for the parameters of the copula $C_{a,b}$.

A simulation study was conducted to assess the performance of the $\left(\tau ,\rho \right)$-inversion method in comparison with bivariate L-moments for estimating the dependence parameters $\left(a,b\right)$ of the radial copula. The evaluation of performance is conducted through the assessment of bias and RMSE, which are defined below:

$\text{Bias}=\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}\left({\hat{\theta }}_i-\theta \right),\text{RMSE}={\left(\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left({\hat{\theta }}_i-\theta \right)}^2\right)}^{1/2},$

where ${\hat{\theta }}_i$ represents an estimate of $\theta$ derived from the $i$-th sample out of $N$ generated samples obtained from the underlying copula [25] . For this study, we set $N=1000$. To evaluate the reduction in bias and RMSE as the sample size increases to $n=50,100,200,500$, the estimation procedures for both methods were repeated by solving the systems in 11 and 15.

The chosen values for the parameters $\left(a,b\right)$ must be reasonable, ensuring that each pair reflects a specific level of dependence: weak, moderate, or strong. Table 1 shows the selected values of the real parameters, while Table 2 gives an overview of the bias and RMSE for both the $\left(\tau ,\rho \right)$-inversion estimator and the BLM estimator applied to the radial copula. The results for bias and RMSE show that the BLM estimator consistently outperforms the $\left(\tau ,\rho \right)$-inversion estimator across all sample sizes. The outperformance of the BLM estimator over the $\left(\tau ,\rho \right)$-inversion method is consistent with the findings of Brahimi et al. [25] . This is because the $\left(\tau ,\rho \right)$-inversion method involves solving an optimization problem under constraints, whereas the BLM estimator directly solves a system of equations, which is generally more efficient and stable for parameter estimation. Furthermore, as the sample size increases from $n=50$ to $n=500$, both the bias and RMSE for the parameters $\left(a,b\right)$ typically decrease, indicating enhanced performance of the estimators with larger samples.

In this section, we investigate the goodness-of-fit performance of radial copulas under different dependence structures. We assess the fit of radial copulas to two datasets previously analyzed in the literature using different bivariate distributions, including various copulas.