2. Tail dependence: one-factor copula model with BB1 family

In the following, we derive the lower tail dependence coefficients for the one-factor copula model with the BB1 family for the linking copula. To this end, we define the BB1 family of copulas indexed by two parameters $\theta> 0$ and $\delta\geq 1$ as that given by

(2.1) \begin{equation}C(u,v;\theta, \delta) = \left\{1+\left[ \left(u^{-\theta }-1\right)^{\delta}+ \left(v^{-{\theta }}-1 \right)^{\delta} \right]^{\frac{1}{\delta} }\right\}^{-\frac{1}{\theta} }, \end{equation}

see Joe and Hu (Reference Joe and Hu1996) and section 5.2 of Joe (Reference Joe1997). This is the copula derived from the bivariate Weibull distribution of Lu and Bhattacharyya (Reference Lu and Bhattacharyya1990). Also, the BB1 copula is referred to as the generalized Clayton copula, see page 259 of McNeil et al. (Reference McNeil, Frey and Embrechts2015).

In Joe and Hu (Reference Joe and Hu1996), it is shown that for an Archimedean copula of the form

(2.2) \begin{equation}C(u,v)=\eta \left(\eta^{-1}(u)+\eta^{-1}(v) \right), \end{equation}

the lower tail dependence coefficient equals $2 \lim_{x \to \infty } \left[\eta'(2x)/ \eta'(x) \right]$ . Since the expression in (2.1) can be written in the form (2.2) with $\eta(x)= \left(1+x^{1/\delta }\right)^{- 1/ \theta } \in (0,1]$ and noting that $\eta^{-1}(x)=\left(x^{-\theta }-1\right)^{\delta}$ , the tail dependence coefficient for a BB1 copula equals

\begin{equation*}2\lim_{x \to \infty } \frac{\eta'(2x)}{\eta'(x)}=2 \lim_{x \to \infty } \left(\frac{1+(2x)^{\frac{1}{\delta} }}{1+x^{\frac{1}{\delta} }} \right)^{- \frac{1}{\theta}-1 } 2^{\frac{1}{\delta}-1 }= 2^{-\frac{1}{ \theta \delta } }.\end{equation*}

The normalized contour plots in Figure 2 display the ability of the BB1 family to model a variety of tail dependence.

Figure 2. Normalized contour plots for BB1 family with parameters $(\theta,\delta)=(0.113, 3.80)$ , $(0.585, 0.171),$ and $(2.63, 1.18)$ . The lower and upper tail dependence coefficients are $(0.2, 0.8)$ , $(0.5, 0.5),$ and $(0.8, 0.2)$ , respectively.

Figure 3. Contour plot for one-factor copula with BB1 family as the linking copula with the same parameters as Figure 2.

Henceforth, we consider the case of a one-factor copula with the linking copulas belonging to the BB1 family with the parameters of the i-th linking copula denoted by $\theta_i(>0),\delta_i(\geq 1)$ , for $i=1,\ldots,d$ . The contour plots in Figure 3 display the ability of the one-factor copula with BB1 family as the linking copula to model a variety of tail dependence. The tail dependence coefficients $\chi_{ij}$ , for $1 \leq i < j \leq d$ , are given by

(2.3) \begin{equation}\chi_{ij}= \lim_{u \downarrow 0}\; \frac{1}{u} \int_0^1 C_{i|0}(u|v) C_{j|0}(u|v) dv=\lim_{u \downarrow 0}\; \int_0^{\frac{1}{u}} C_{i|0}(u|uw) C_{j|0}(u|uw) dw, \end{equation}

where

(2.4) \begin{align}C_{i|0}(u|uw) &=\left\{\left[ \frac{u^{-\theta_i}-1}{(uw)^{-\theta_i}-1} \right]^{\delta_i}+1\right\}^{\frac{1}{\delta_i}-1}\nonumber\\[5pt] & \qquad \cdot \left\{ (uw)^{\theta_i}+\left[\left[w^{\theta_i}-(uw)^{\theta_i}\right]^{\delta_i}+ \left[1-(uw)^{\theta_i}\right]^{\delta_i} \right]^{\frac{1}{\delta_i}} \right\}^{-\frac{1}{\theta_i}-1}\mkern-45mu. \end{align}

Theorem 2.1. The tail dependence coefficients $\chi_{ij}$ , for $1 \leq i < j \leq d$ , of the one-factor copula with the linking copula being the Family BB1 are given by

\[\chi_{ij}= \int_0 ^ \infty \left(1+ w^{\theta_i \delta_i} \right)^{-\frac{1}{\theta_i \delta_i}-1} \left(1+ w^{\theta_j \delta_j} \right)^{-\frac{1}{\theta_j \delta_j}-1} dw, \quad \theta_i,\theta_j\in(0,\infty),\; \delta_i,\delta_j\geq 1.\]

In the case that $\theta_i=\infty$ , and $\theta_j\in(0,\infty)$ , we have

$$ \chi_{ij}=\int_0^1 (1+w^{\theta_j \delta_j})^{-\frac{1}{\theta_j \delta_j}-1} dw= 2^{-\frac{1}{\theta_j\delta_j}}.$$

Finally, if $\theta_i=\theta_j=\infty$ , $\chi_{ij}=1$ .

Proof. The case when either or both $\theta$ ’s equal infinity, the result follows from the fact that the BB1 copula with $\theta=\infty$ is the comonotonic copula. Hence, in this case, the corresponding coordinate(s) equals the common factor, and $\chi_{ij}$ corresponds to that for the BB1 copula. For the case when $\theta_i,\theta_j\in(0,\infty)$ , note that the limits of the conditional distributions $C_{i|0}(u|uw)$ , for $i =1,\ldots,d$ , as u approaches 0, are given by

(2.5) \begin{equation}\lim_{u \downarrow 0}\; C_{i|0}(u|uw) =\left(1+w^{\theta_i \delta_i}\right)^{\frac{1}{\delta_i} -1+\frac{1}{\delta_i} \left(-\frac{1}{\theta_i} -1\right)} =\left( 1+w^{\theta_i \delta_i} \right)^{-\frac{1}{\theta_i \delta_i}-1}, \quad i=1,\ldots,d.\end{equation}

The result now follows by applying the dominated convergence theorem, as the integrand

\[w\mapsto C_{i|0}(u|uw) C_{j|0}(u|uw) I{_{(0<uw<1)}}\]

is bounded for all $u\in(0,1)$ by the integrable $g_ig_j$ of Lemma A.1 in the appendix.

Note that the tail dependence coefficients $\chi_{ij}$ , can be expressed as $\chi_{ij}=\Psi (\theta_i \delta_i,\theta_j \delta_j)$ , for $1 \leq i < j \leq d$ , where for $0<x,y<\infty$ ,

(2.6) \begin{align} \begin{split} \Psi (x,y)&\;:\!=\;\int_0 ^ \infty \left(1+ w^x \right)^{-\frac{1}{x}-1} \left(1+ w^y \right)^{-\frac{1}{y}-1} dw,\\[5pt] \Psi (x,\infty)&\;:\!=\;\Psi(\infty,x)\;:\!=\;\Psi_1(x)\;:\!=\;\int_0^1 (1+w^x)^{-\frac{1}{x}-1} dw=2^{-\frac{1}{x}}, \\[5pt] \Psi(\infty,\infty)&\;:\!=\;1. \end{split}\end{align}

Hence, the study of the geometric properties of the set of TDMs generated by the one-factor copula with the BB1 family for linking copulas, requires study of the symmetric function $\Psi$ . Toward this, we denote the integrand above by $\psi$ , that is

\begin{eqnarray*} \psi (x,y,w)\;:\!=\;\left(1+w^x \right)^{-\frac{1}{x}-1}\left(1+w^y \right)^{-\frac{1}{y}-1},\qquad x,y>0,\; w \geq 0.\end{eqnarray*}

The proof of the following proposition is in the appendix.

The following corollary conveniently lists some properties of $\Psi$ that are needed for the results in the following sections, and the proof can be found in the appendix.

3. Representativeness of $\mathcal{T}_d'$ as a subset of $\mathcal{T}_d$

To study the geometric properties of $\mathcal{T}_d'$ , we define the embedding function $\mathbf{\Psi}_d\;:\; \mathbb{R}_+^d \to \mathcal{T}_d' $ by

\[\left(\mathbf{\Psi}_d (\mathbf{x}) \right)_{ij} \;:\!=\; \begin{cases} \Psi(x_{i},x_{j}), & 1 \leq i\neq j \leq d;\\[5pt] 1, & i=j,\end{cases}\]

where $ \mathbf{x}=[x_1,x_2,\ldots,x_d] \in \mathbb{R}_+^d$ is the vector of the parameters defining the tail dependence coefficients, that is $x_i\;:\!=\;\theta_i\delta_i$ , for $i=1,\ldots,d$ . Note that $\mathcal{T}_d'$ is the range of $\mathbf{\Psi}_d(\mathbb{R}_+^d)$ .

For $d=2$ , Corollary 2.1 readily shows that $\mathcal{T}_2'=(0,1] \subset \mathcal{T}_2=[0,1]$ . On the other hand, when $d \geq 3$ , $\mathcal{T}_d'$ is a strict subset of $\mathcal{T}_d$ . To see this, we consider MA(1) matrices with unit diagonal, that is, $\mathbf{T}=(t_{ij})$ , where $t_{ij}=I_{i=j}+\alpha I_{|i-j|=1}$ , for $1 \leq i,j \leq d$ and $\alpha \in (0, 1/2]$ . Any such matrix is a member of $\mathcal{T}_d$ , for $d \geq 2$ (see Section 4.1 of Embrechts et al., Reference Embrechts, Hofert and Wang2016 or Proposition 6 of Shyamalkumar and Tao, Reference Shyamalkumar and Tao2020). However, its third-order leading principal sub-matrix of the form

\[\begin{bmatrix}1 & \alpha& 0 \\[5pt] \alpha & 1 & \alpha \\[5pt] 0 & \alpha & 1\end{bmatrix}, \quad \alpha \in (0, 1/2].\]

is not feasible by Proposition 3.2 below. Therefore, $\mathcal{T}_d \setminus \mathcal{T}_d'$ in non-empty.

In the first subsection, we establish some topological and geometric characteristics of $\mathcal{T}_3'$ . The second subsection introduces an imposed ordering and elaborates on geometric properties for $\mathcal{T}_d'$ , $d \geq 4$ .

3.1. Volume of $\mathcal{T}_d'$ relative to $\mathcal{T}_d$

As a starting point, we aim to determine the precise volume of the TDMs generated by the one-factor copula, using the Family BB1 as the linking copula in a three-dimensional setting. Our analysis begins with an exploration of a key property concerning the three-dimensional tail dependence coefficients, which is presented in the following proposition.

Proposition 3.1. Let $\chi_{12},\chi_{13},\chi_{23} \in (0,1]$ . Then,

\begin{eqnarray*}\chi_{12}=\Psi (x,y), \quad \chi_{13}=\Psi (x,z), \quad \hbox{and} \quad \chi_{23}=\Psi (y,z),\end{eqnarray*}

can have at most one solution. Furthermore, if $\{x,y,z\}$ is the solution, then $\chi_{12} \leq \chi_{13} \leq \chi_{23}$ if an only if $x\leq y \leq z$ .

Proof. Suppose that $\{x,y,z\}$ and $\{x',y',z'\}$ are both solutions to the system of equations. Without loss of generality, we can assume that $x \geq x'$ . By Corollary 2.1 (ii), we know that $y \leq y'$ and $z \leq z'$ from the first two equations. The third equation shows that $y=y'$ and $z=z'$ . Hence, $x=x'$ , and the two solutions are identical.

Now, let $\{x,y,z\}$ be the solution. The if part follows from monotonicity of $\Psi$ from Proposition 2.1. For the only if part, let $\chi_{12} \leq \chi_{13} \leq \chi_{23}$ . Again using monotonicity of $\Psi$ , the first two equations along with $\chi_{12} \leq \chi_{13}$ indicate that $y \leq z$ , and the last two equations similarly imply that $x \leq y$ . Therefore, we have $x\leq y \leq z$ .

Before delving into properties of ${\mathcal{T}}_d'$ at higher dimensions, we further examine the representativeness of ${\mathcal{T}}_3'$ . Proposition 3.1 establishes that $\mathcal{T}_3'$ displays invariance with respect to permutation of the coordinates. For expositional ease, we hence focus on the subset of the TDMs satisfying $\chi_{12} \leq \chi_{13} \leq \chi_{23}$ . This subset, which we denote as $ \mathcal{O}_3$ , takes the form of a tetrahedron with vertices at (0,0,0), (0,0,1), $(0, 1/2, 1/2)$ , and (1,1,1). We further denote the collection of feasible points within $\mathcal{O}_3$ by $\mathcal{O}'_{\!\!3}$ , that is $\mathcal{O}'_{\!\!3} \;:\!=\;\mathcal{T}_3' \cap \mathcal{O}_3$ . The volume of $\mathcal{O}_3$ (resp. $\mathcal{O}'_{\!\!3}$ ) is clearly one-sixth that of $\mathcal{T}_3$ (resp. $\mathcal{T}_3'$ ). Figure 4(a) displays the six cells of $\mathcal{T}_3$ belonging to its partition induced by the ordering of the coordinates. The following proposition gives a necessary and sufficient condition for membership in ${\mathcal{T}}_3'$ , and using it derives its volume.

Proposition 3.2. A matrix of the form

\[\begin{bmatrix} 1 & a & b\\[5pt] a & 1 & c\\[5pt] b & c & 1\end{bmatrix}, \quad 0 < a \leq b \leq c \leq 1,\]

is in $\mathcal{T}'_{\!\!3}$ if and only if $a \geq \Psi \left(\Psi_1 ^{-1}(b),\Psi_1 ^{-1}(c) \right)$ . Furthermore, the $\mathcal{T}'_{\!\!3}$ has a volume of about $0.12882$ , which is about 26% of the volume of ${\mathcal{T}}_3$ .

Proof. For the only if part, by Proposition 2.1, we establish that for any given $b=\Psi(x,z)$ and $c=\Psi(y,z)$ , where $0 < b \leq c \leq 1$ , the lower bounds for x and y are determined as $\Psi_1 ^{-1}(b)$ and $\Psi_1 ^{-1}(c)$ , respectively. Consequently, the lower bound for $a=\Psi(x,y)$ is expressed as $\Psi \left(\Psi_1 ^{-1}(b),\Psi_1 ^{-1}(c) \right).$

Figure 4. $\mathcal{O}_3$ and the lower bound on $\chi_{12}$ within $\mathcal{O}'_{\!\!3}$ .

For the if part, let a, b, and c satisfy $\Psi(\Psi_1^{-1}(b), \Psi_1^{-1}(c)) \leq a \leq b \leq c \leq 1$ . From Proposition 2.1, we have $\Psi$ is a symmetric, strictly increasing, and continuous function on $(0,\infty]^2$ . Since,

$$ \Psi(\Psi_1^{-1}(b), 0) = 0, \hbox{ and } \Psi(\Psi_1^{-1}(b), \infty) = b,$$

using continuity of $\Psi$ , we have the existence of a $z \geq \Psi_1^{-1}(c)$ satisfying $a = \Psi(\Psi_1^{-1}(b), z)$ . This establishes that the parameter $(\Psi_1^{-1}(b), z, \infty)$ corresponds to the TDM with off-diagonal elements of a, b, and $\Psi(z, \infty)$ . It is noteworthy that $\Psi(z, \infty) \geq \Psi(\Psi_1^{-1}(c), \infty) = c$ . If $\Psi(z, \infty) = c$ , our proof is complete; we now focus on the remaining case of $c < \Psi(z, \infty) \leq 1$ . Consider functions $\eta_{a,b}\;:\; (0,\infty) \to (\Psi_1^{-1}(a),z)$ and $\eta_{b,b}\;:\; (0,\infty) \to (\Psi_1 ^{-1}(b),\infty)$ , defined by

\[a=\Psi(\Psi_1 ^{-1}(b)+x, \eta_{a,b}(x)), \hbox{ and } b=\Psi(\Psi_1 ^{-1}(b)+x, \eta_{b,b}(x)),\]

respectively. Given that $\Psi$ is strictly increasing and smooth, both $\eta_{a,b}$ and $\eta_{b,b}$ are continuous (see Lemma A.3 in the Appendix) and strictly decreasing. Defining $\phi(\!\cdot\!)$ by

\[\phi(x)\;:\!=\;\Psi(\eta_{a,b}(x),\eta_{b,b}(x)), \quad x \in (0,\infty),\]

we note that

\[\lim_{x \downarrow 0}\phi(x)= \Psi(z, \infty) > c, \hbox{ and } \lim_{x \uparrow \infty}\phi(x)=\Psi \left(\Psi_1 ^{-1}(a),\Psi_1^{-1}(b) \right) < a \leq c.\]

The continuity of $\Psi(\cdot,\cdot)$ , $\eta_{a,b}$ and $\eta_{b,b}$ , imply the continuity of $\phi$ , and hence we have the existence of $z' > 0$ satisfying $\phi(z')=\Psi(\eta_{a,b}(z'),\eta_{b,b}(z'))=c$ . Consequently, the TDM with off-diagonal elements a, b, and c corresponds to the parameters $(\Psi_1 ^{-1}(b)+z', \eta_{a,b}(z'), \eta_{b,b}(z'))$ .

Recalling that $\mathcal{O}'_{\!\!3}$ is given by

\[\left\{(\Psi(x,y), \Psi(x,z), \Psi(y,z) ), \quad \text{for } 0 < x \leq y \leq z\right\},\]

the curved surface of the lower bound on $\chi_{12}$ is plotted in Figure 4(b). The region above this curved surface represents $\mathcal{O}'_{\!\!3}$ . Through integration across the curved surface and the $\chi_{13}-\chi_{23}$ plane, we can compute the volume below the surface as:

\[\int_0^1 \int_0^c \Psi \left(\Psi_1 ^{-1}(b),\Psi_1 ^{-1}(c) \right) db \; dc \approx 0.145197.\]

As a result, the volume of $\mathcal{O}'_{\!\!3}$ is approximately $0.021470$ , and this result, when multiplied by six, yields the volume of $\mathcal{T}'_{\!\!3}$ .

Remark 3.1. We observe that $\mathcal{O}'_{\!\!3}$ is not a convex set. This is so as the midpoint of the line segment between the points (0,0,1) and $(1/3,1/2,1/2)$ of $\mathcal{O}'_{\!\!3}$ (i.e. $(1/6,1/4,3/4)$ ) does not belong to $\mathcal{O}'_{\!\!3}$ . This is verified by referring to Proposition 3.2 and noting that:

\[\Psi \left(\Psi_1 ^{-1}(3/4),\Psi_1 ^{-1}(1/4) \right)= 0.23 > 1/6.\]

Furthermore, note that the non-convexity of $\mathcal{O}'_{\!\!3}$ implies the non-convexity of $\mathcal{T}'_{\!\!3}$ .

Theorem 3.1. $\mathcal{T}'_{\!\!d}$ has a non-zero volume in $\mathbb{R}_+^{\binom{d}{2}}$ if and only if $d \leq 3$ .

Proof. Recall that the tail dependence coefficients of a d-dimensional one-factor copula with the linking copulas from a BB1 family are a function of the d-dimensional vector $(\theta_1\delta_1, \ldots, \theta_d\delta_d)\in(0,\infty)^d$ . Defining,

\begin{eqnarray*} \phi\;:\; (0,\infty)^d &\longrightarrow & [0,1]^{\binom{d}{2}}\\[5pt] (x_1,\ldots,x_d) &\longmapsto & \left(\Psi(x_1,x_2), \Psi(x_1,x_3),\ldots,\Psi(x_{d-1},x_d) \right),\end{eqnarray*}

we see that the interior of $\mathcal{T}'_{\!\!d}$ is contained in the range of $\phi$ . Note that $d < d(d-1)/2$ , for $d \geq 4$ , hence the domain of $\phi$ is in a lower dimensional Euclidean space than its range. Moreover, Proposition 2.1 implies that $\phi$ is in $C^1$ . Hence, by using Sard’s theorem (see Theorem A.1 of the Appendix), we see that the range of $\phi$ has zero Lebesgue measure in $[0,1]^{\binom{d}{2}}$ .

Remark 3.2. It is important to note that choosing a different linking copula from the BB families listed in Joe (Reference Joe1997) would lead to a different set of TDMs generated by the one-factor copula for $d=3$ . In Table A1 in the Appendix, we provide additional information on related functions and the volume of the one-factor copula using two other BB families (BB4 and BB7) suggested in Section 8.6 of Joe (Reference Joe2011). To calculate the volume for the BB4 family, we employed a hit-and-run sampler, a method detailed in Kroese et al. (Reference Kroese, Taimre and Botev2013) and Shyamalkumar and Tao (Reference Shyamalkumar and Tao2022). This approach allowed us to uniformly generate a sample of 100,000 points from $\mathcal{T}_3$ , which we then assessed for their feasibility within the BB4 family. Importantly, it is worth noting that the tail dependence coefficient of the BB7 family adheres to the same mathematical form as that of the BB1 family. As a result, both families share identical volumes.

3.2. An imposed ordering

In the following, we show that the one-factor structure imposes a rather restrictive partial ordering on the tail dependence coefficients. While the discussion below is in the case of a BB1 linking copula, the development suggests that the phenomenon is more general. We begin by showing that, in general, on the contrary, all possible orderings of the tail dependence coefficients are possible. In the following ${\mathcal{I}}_d\;:\!=\;\{(1,2), (1,3),\ldots,(d-1,d)\}$ , the set of all ordered pairs of positive integers less than or equal to d.

Theorem 3.2. Given a linear ordering $\prec$ on ${\mathcal{I}}_d$ , there exists a TDM, $\mathbf{T}_d=(t_{ij})$ , such that the mapping $(i,j)\in{\mathcal{I}}_d\mapsto t_{ij}$ is strictly increasing in $\prec$ .

Proof. For $(i,j)\in{\mathcal{I}}_d$ , let $0\leq \varepsilon_{ij} \leq 1/d$ be such that $(i,j)\mapsto \varepsilon_{ij}$ is strictly decreasing in $\prec$ , and define $\varepsilon_{ji}\;:\!=\;\varepsilon_{ij}$ . Defining $t_{ij}\;:\!=\; 1/d-\varepsilon_{ij}$ , for $1 \leq i <j \leq d$ , we see that the mapping $(i,j)\mapsto t_{ij}$ is strictly increasing in $\prec$ . In the following, we will show that $\mathbf{T}_d=(t_{ij})$ is a TDM. By Theorem 1 of Shyamalkumar and Tao (Reference Shyamalkumar and Tao2020), it is equivalent to show that $\mathbf{T}_d/d$ is a Bernoulli Compatible Matrix (BCM). Also, by Corollary 3 therein, this is equivalent to showing the existence of events $A_i$ , $i=1,\ldots d$ , on a probability space with identical probabilities of $1/d$ such that $t_{ij}=\textrm{Pr}\left(A_i|A_j\right)$ . Toward this, we construct disjoint sets $A_{ij}$ , for $1 \leq i <j \leq d$ , and $B_i$ , $i=1,\ldots d$ , on a non-atomic probability space with probabilities

\[\textrm{Pr}\left(A_{ij}\right)=\frac{1}{d} \left(\frac{1}{d}-\varepsilon_{ij} \right), \quad \hbox{and} \quad \textrm{Pr}\left(B_i\right)=\frac{1}{d}-\sum_{j \neq i}\left(\frac{1}{d^2}-\frac{\varepsilon_{ij}}{d} \right).\]

This can be done as the sum of the probabilities of the above-mentioned sets are

\begin{align*}\sum_{1 \leq i <j \leq d}\textrm{Pr}\left(A_{ij}\right)+\sum_{i=1}^d \textrm{Pr}\left(B_i\right)&= \sum_{1 \leq i <j \leq d} \frac{1}{d} \left(\frac{1}{d}-\varepsilon_{ij} \right)+ \sum_{ i=1}^d \left[\frac{1}{d}-\sum_{j \neq i}\left(\frac{1}{d^2}-\frac{\varepsilon_{ij}}{d}\right) \right]\\[5pt] &=1-\sum_{1 \leq i <j \leq d} \frac{1}{d} \left(\frac{1}{d}-\varepsilon_{ij} \right)\leq1.\end{align*}

With $A_{ji}\;:\!=\;A_{ij}$ , for $1 \leq i <j \leq d$ , and

\[A_i \;:\!=\; B_i \operatorname{{\cup}\!\!\!{\cdot}\,} \; \left(\operatorname{{\bigcup}\!\!\!\!{\cdot}\,}_{j \neq i} A_{ij}\right), \quad \text{for} \; i=1,\ldots,d,\]

we see that $\textrm{Pr}\left(A_i\right)=1/d$ , $A_i\cap A_j=A_{ij}$ , and hence $t_{ij}=\textrm{Pr}\left(A_i| A_j\right)$ satisfies the sought ordering.

Remark 3.3. From the above, it is easily checked that for $\Vert \varepsilon\Vert_2\leq \sqrt{2}/d^2$ , by Cauchy-Schwartz inequality we have $\Vert \varepsilon\Vert_1\leq 1/d$ . The latter suffices for the validity of the above construction, and hence the above proof implies further that the TDM with off-diagonal elements equal to $1/d$ is in the interior of ${\mathcal{T}}_d$ , and moreover using the fact that ${\mathcal{T}}_d$ is a convex polytope we have that the line segment between the extreme points origin and $\mathbf{1}$ of ${\mathcal{T}}_d$ lies in its interior.

In the following, we assume that the linking copula is the BB1 copula and define $\eta_i\;:\!=\;\theta_i \delta_i$ , $i=1,\ldots,d$ . In the case $d=3$ , we have the ordering of $\eta_i$ ’s specifying the ordering of the tail dependence coefficients by the monotonicity of $\Psi(\cdot,\cdot)$ . In particular, if $\eta_1\leq \eta_2\leq \eta_3$ , then we have $\chi_{12}\leq \chi_{13}\leq \chi_{23}$ . For $d\geq 3$ , the lemma below shows that the ordering of $\eta_i$ s specifies a partial ordering on ordered pairs (i,j), for $1\leq i<j\leq d$ , denoted by $\prec_\eta$ , and that the mapping $(i,j)\mapsto \chi_{ij}$ is non-decreasing with respect to this partial order. Figure 5(a) contains the Hasse diagram for the partial ordering with non-decreasing $\eta_i$ s for $d=5$ and specifies the partial ordering for $d\leq5$ .

Figure 5. Linear ordering of the poset: $d=5$ ; increasing $\eta_i$ s.

For the following development, we require an arbitrarily chosen strict total order on ${\mathcal{I}}_d$ . For simplicity, we use the reverse lexicographic ordering and denote it by $\prec_{\scriptstyle rl}$ . The notation used below does suppress the dependence on $\prec_{\scriptstyle rl}$ as we note below that it has little impact on the main conclusion.

Proposition 3.3. Given $\eta_i$ s, we define $(i,j)\prec_\eta(k,l)$ for $(i,j),(k,l)\in{\mathcal{I}}_d$ if $\min(\eta_i,\eta_j)\leq \min(\eta_k,\eta_l)$ and $\max(\eta_i,\eta_j)\leq \max(\eta_k,\eta_l)$ , with ties broken using $\prec_{\scriptstyle rl}$ . The binary relation $\prec_\eta$ is a partial order, and moreover, the mapping $(i,j)\mapsto \chi_{ij}$ is non-decreasing with respect to $\prec_\eta$ .

Proof. To show that $\prec_\eta$ is a partial order, we need to demonstrate the following properties for any (i,j), $(k,l),(u,v)\in {\mathcal{I}}_d$ :

1. i. Reflexivity: $(i,j)\prec_\eta(i,j)$ ;

2. ii. Antisymmetry: If $(i,j)\prec_\eta(k,l)$ and $(k,l)\prec_\eta(i,j)$ , then $(i,j)=(k,l)$ ;

3. iii. Transitivity: If $(i,j)\prec_\eta(k,l)$ and $(k,l)\prec_\eta(u,v)$ , then $(i,j)\prec_\eta(u,v)$ .

Part i. follows from the definition of $\prec_\eta$ and that $\prec_{\scriptstyle rl}$ is a partial order. For part ii., we have that

\[\min(\eta_i,\eta_j)\leq \min(\eta_k,\eta_l) \leq \min(\eta_i,\eta_j) \quad \text{and} \quad \max(\eta_i,\eta_j)\leq \max(\eta_k,\eta_l) \leq \max(\eta_i,\eta_j).\]

Hence, $\min(\eta_i,\eta_j)=\min(\eta_k,\eta_l)$ and $\max(\eta_i,\eta_j)=\max(\eta_k,\eta_l)$ . Since ties are broken (using $\prec_{\scriptstyle rl}$ ), we have $(i,j)=(k,l)$ . For part iii., the ordering implies that

\[\min(\eta_i,\eta_j)\leq \min(\eta_k,\eta_l) \leq \min(\eta_u,\eta_v) \quad \text{and} \quad \max(\eta_i,\eta_j)\leq \max(\eta_k,\eta_l) \leq \max(\eta_u,\eta_v).\]

Hence, since ties are broken using $\prec_{\scriptstyle rl}$ , we have $(i,j)\prec_\eta(u,v)$ .

The non-decreasing nature of the mapping $(i,j)\mapsto \chi_{ij}$ with respect to $\prec_\eta$ follows from Proposition 2.1 as for $(i,j)\prec_\eta(k,l)$ we have that

\begin{align*}\chi_{ij}=\Psi(\eta_i,\eta_j) &=\Psi(\min(\eta_i,\eta_j),\max(\eta_i,\eta_j)) \\[5pt] &\leq \Psi(\min(\eta_k,\eta_l),\max(\eta_k,\eta_l))=\Psi(\eta_k,\eta_l)=\chi_{kl}.\end{align*}

On the other hand, given a set of tail dependent coefficients corresponding to a TDM in ${\mathcal{T}}_d'$ , we can infer the ordering of the underlying $\eta_i$ s as demonstrated by the below proposition.

Proposition 3.4. Let $\bar{\chi}\;:\!=\;(\chi_{1,2},\ldots,\chi_{d-1,d})$ be tail dependence coefficients corresponding to a member of $\mathcal{T}_d'$ , with $d \geq 3$ . Also, let $n_{\bar{\chi}}$ be the function defined on $\{1,\ldots,d\}$ by

\begin{align*}n_{\bar{\chi}}(i) &=\sum_{m=1}^{d(d-1)/2} 2^{m-1} I_{(i_m=i \text{ or } j_m=i)},\end{align*}

where $(i_m,j_m) \in \mathcal{I}_d$ , for $m=1,\ldots,d(d-1)/2$ , is defined by $\chi_{i_1,j_1} \leq \cdots \leq \chi_{i_{d(d-1)/2},j_{d(d-1)/2}}$ , with the ties broken using $\prec_{\scriptstyle rl}$ . Then $n_{\bar{\chi}}$ is injective, and $\eta_i$ s underlying $\bar{\chi}$ are non-decreasing with respect to $<_\chi$ , the strict total order on $\{1,\dots,d\}$ under which $n_{\bar{\chi}}$ increasing.

Proof. To show that $n_{\bar{\chi}}$ is injective, assume on the contrary that $n_{\bar{\chi}}(i) = n_{\bar{\chi}}(j)$ , for some $i<j$ . Then,

\[\sum_{m=1}^{d(d-1)/2} 2^{m-1} I_{(i_m=i \text{ or } j_m=i)}=\sum_{m=1}^{d(d-1)/2} 2^{m-1} I_{(i_m=j \text{ or } j_m=j)},\]

implying that $I_{(i_m=i \text{ or } j_m=i)} = I_{(i_m=j \text{ or } j_m=j)}$ , for $m=1,\ldots,d(d-1)/2$ . Since the latter implies that $i=j$ , we have a contradiction. Thus, $n_{\bar{\chi}}$ is injective.

Next, we prove that the $\eta_i$ s underlying $\bar{\chi}$ are non-decreasing with respect to the total order $<_\chi$ . For $i<j$ in the total order $<_\chi$ , we have $n_{\bar{\chi}}(i) < n_{\bar{\chi}}(j)$ , that is,

\[\sum_{m=1}^{d(d-1)/2} 2^{m-1} I_{(i_m=i \text{ or } j_m=i)}< \sum_{m=1}^{d(d-1)/2} 2^{m-1} I_{(i_m=j \text{ or } j_m=j)}.\]

Hence, $\arg\max_m(i_m=i \text{ or } j_m=i) \leq \arg\max_m(i_m=j \text{ or } j_m=j)$ . Now, consider two cases. First, if $\arg \max_m(i_m=i \text{ or } j_m=i) < \arg\max_m(i_m=j \text{ or } j_m=j)$ , then $\chi_{ki} \leq \chi_{k'i}\leq \chi_{kj}$ , where $\chi_{kj}$ is the largest value in $\bar{\chi}$ that has j as an index, and $\chi_{k'i}$ is the largest value in $\bar{\chi}$ that has i as an index. By Proposition 2.1, this implies $\eta_i \leq \eta_j$ . Second, if $\arg\max_m(i_m=i \text{ or } j_m=i) = \arg \max_m(i_m=j \text{ or } j_m=j) \;:\!=\; l$ , then $\arg\max_{m\neq l}(i_m=i \text{ or } j_m=i) < \arg \max_{m\neq l}(i_m=j \text{ or } j_m=j)$ . This means that $\chi_{ki} \leq \chi_{k'i}\leq \chi_{kj}$ , where $\chi_{kj}$ is the second largest value in $\bar{\chi}$ that has j as an index, and $\chi_{k'i}$ is the second largest value in $\bar{\chi}$ that has i as an index. Hence, by Proposition 2.1, we again have $\eta_i \leq \eta_j$ . In both cases, we have shown that for $i<j$ in the total order $<_\chi$ , $\eta_i \leq \eta_j$ . Therefore, the $\eta_i$ s underlying $\bar{\chi}$ are non-decreasing with respect to the total order $<_\chi$ .

Proposition 3.4 implies that $\mathcal{T}_d'$ can be partitioned into $d!$ subsets with each corresponding to a permutation of $\{1,\ldots,d\}$ , consistent with the ordering of the underlying $\eta_i$ s. Note that the use of $\prec_{\scriptstyle rl}$ only is relevant for $\bar{\chi}$ with some tail dependence coefficients assuming the same value. By Proposition 3.3 we see that each of these subsets can be further partitioned corresponding to a permutation of $\mathcal{I}_d$ , consistent with the ordering of the tail dependence coefficients, and by symmetry we see that there are $(d(d-1)/2)!/d!$ such subsets. In the case $d=4$ , we have 30 such cells of $\mathcal{T}_4$ corresponding to the identity permutation on $\{1,\ldots,d\}$ , denoted by $\mathcal{O}_4$ , corresponding to an increasing $\eta_i$ s. Interestingly, only two of them are nonempty if we ignore $\bar{\chi}$ with some tail dependence coefficients assuming the same value. These correspond to the permutations,

\[((1,2),(1,3),(2,3),(1,4),(2,4),(3,4)) \;\hbox{and}\;((1,2),(1,3),(1,4),(2,3),(2,4),(3,4)),\]

consistent, respectively, with

\[\chi_{12}<\chi_{13}<\chi_{23}<\chi_{14}<\chi_{24}<\chi_{34}, \;\hbox{and}\; \chi_{12}<\chi_{13}<\chi_{14}<\chi_{23}<\chi_{24}<\chi_{34}.\]

This is so as these are the only permutations corresponding to linear orderings consistent with the partial order of Figure 5(a). In other words, of the 720 (non-empty interiors by Theorem 3.2 and Remark 3.3) cells of ${\mathcal{T}}_4$ determined by the ordering $<_\chi$ , only a mere $48\; (2\times 4!)$ have a non-empty intersection of their interiors with ${\mathcal{T}}_4'$ . The below theorem generalizes this observation for all $d\geq 3$ . In Table A2 of the Appendix, the volumes of these cells of $\mathcal{O}_4$ calculated using polymake (see Gawrilow and Joswig, Reference Gawrilow and Joswig2000) are given. It is worth noting that the nonempty cells 1 and 7 – polytopes – share a facet of dimension 5. The adjacency structure for all 30 polytopes of $\mathcal{O}_4$ is depicted in Figure A1 in the Appendix, where polytopes connected with lines indicate that they share a common facet of dimension 5.

Remark 3.4. For any point in $\mathcal{O}_4$ , we can find the lower bound of its Euclidean distance to $\mathcal{O}_4'$ , the feasible area under the strict total order $<_\chi$ , by projecting it onto cells 1 or 7 of the partition $\mathcal{O}_4$ . For example, the lower bound from the vertex $(1,0,0,0,0,1)\in \mathcal{O}_4$ to cells 1 and 7 is $\sqrt{0.8}$ . The lower bound to the feasible area can be considered as a lower bound to the closest feasible matrix, which we will discuss in the next section.

For the proof of the theorem, we need the concept of a Young tableaux. A Young tableaux, see Section 5.1.4 of Knuth (Reference Knuth1998), is an arrangement of $n_1\geq n_2\geq \ldots \geq n_{m-1}\geq n_m>0$ distinct integers in an array of left-justified rows with the i-th row containing $n_i$ elements, $i=1, \ldots m$ , such that the elements in each row are in an increasing order (left to right) and the elements in each column are increasing top to bottom. Below, find it convenient to refer to the underlying table structure as a Young graph, refer to the assignment of numbers to the graph as the labeling, and a labeling satisfying the above monotonicity property as a regular labeling. In Table 1a we depict a Young tableaux corresponding to the (4,2,1) partition of 7. In Table 1b, we depict a certain version of a shifted Young tableaux in which the i-th row begins at the i-th column, and in Table 1c, we depict the transpose of the shifted Young tableaux which we will refer to as a Thrall tableaux.

Table 1. Young and Thrall tableaux

While there is an obvious equivalence between a regular labelings of a Thrall graph and its corresponding shifted Young graph, and such labelings yield a regular labeling of the corresponding Young graph, Table 2 shows that not all regular labelings of a Young graph yield a regular labeling of its corresponding Thrall graph. The following result from Thrall (Reference Thrall1952) gives the number of regular labelings of a Thrall graph.

Table 2. Not all Young tableaux correspond to a Thrall Tableaux.

Theorem 3.3 (Thrall Reference Thrall1952). The number of regular labelings of a Thrall graph associated with $\tilde{n}=(n_1,\ldots,n_m)$ , a partition of n, with label set $\{1,\ldots,n\}$ is given by

(3.1) \begin{equation}\frac{n!\Delta(\tilde{n})}{n_1!n_2! \cdots n_m! \nabla(n_1,\ldots, n_m)},\end{equation}

where $n=n_1+\cdots+n_m$ ,

\[\Delta(\tilde{n})\;:\!=\; \prod_{1\leq i < j\leq m} (n_i-n_j), \quad \hbox{and} \quad \nabla(\tilde{n})\;:\!=\; \prod_{1\leq i < j\leq m} (n_i+n_j).\]

The number of Young tableaux of shape $(n_1,\ldots,n_m)$ is given by, see (34) on page 58 of of Knuth (Reference Knuth1998),

(3.2) \begin{equation}\frac{n!\Delta(n_1+m-1,n_2+m-2, \ldots, n_m)}{(n_1+m-1)!(n_2+m-2)! \cdots n_m!}.\end{equation}

Interestingly, the formula in (3.2) was first derived from a group theoretic perspective in Frobenius (Reference Frobenius1900), and later a combinatorial proof was given in MacMahon (Reference MacMahon1909).

Theorem 3.4. For $d\geq 3$ , of the $\binom{d}{2}!$ subsets of ${\mathcal{T}}_d$ determined by $<_\chi$ , the number with nonempty intersection of their interiors with ${\mathcal{T}}_d'$ is bounded above by

(3.3) \begin{equation}N(d)\;:\!=\;d!\binom{d}{2}!\frac{\prod_{i=1}^{d-2} i!}{\prod_{i=1}^{d-1} (2i-1)!}.\end{equation}

In particular, the corresponding proportion decays super-exponentially to zero as

(3.4) \begin{equation}\frac{d\prod_{i=1}^{d-1} i!}{\prod_{i=1}^{d-1} (2i-1)!}\leq e^{-\Omega(d^2)}.\end{equation}

Proof. Let us consider, without loss of generality, the interior of the subset of ${\mathcal{T}}_d$ , which corresponds via $<_\chi$ to the identity permutation on $\{1,\ldots,d\}$ . By Proposition 3.4, points in this interior belonging to ${{\mathcal{T}}}_d'$ have a strictly increasing underlying $\eta_i$ s. Hence, by Propositions 3.3 and 3.1, we have that any such point is strictly increasing in a strict linear order of $\mathcal{I}_d$ that is consistent with the partial order implied by a strictly increasing underlying $\eta_i$ s. This partial order for $d=5$ is depicted in Figure 5(a). So the number of distinct strict total orders corresponding to such points is bounded above by strict linear orders on $\mathcal{I}_d$ consistent with the partial order implied by a strictly increasing underlying $\eta_i$ s. So to prove (3.3), by symmetry with respect to the permutation implied by $<_\chi$ , it suffices to show that the latter equals $N(d)/d!$ .

Considering the case $d=5$ , we see that the number of strict total orders satisfying the partial order of Figure 5(a) equals the number of regular labelings with $\{1,\ldots,10\}$ of a Thrall graph such as that shown in Table 2(c). In general, it should be now clear that we seek the number of regular labelings of a Thrall graph associated with $\tilde{n}_d=(d-1,d-2,\ldots,1)$ , a partition of $n_d=d(d-1)/2$ , with label set $\{1,\ldots,n_d\}$ . The proof of (3.3) is completed by an application of Theorem 3.3, and by observing that

\[\Delta(\tilde{n}_d)=\prod_{i=1}^{d-2} i!, \quad \hbox{and} \quad \nabla(\tilde{n}_d)= \prod_{i=1}^{d-1} \frac{(2i-1)!}{i!}.\]

The super-exponential decay conveyed by (3.4) follows from Lemma A.5 in the Appendix.

While Theorem 3.4 provides an upper bound, it is unlikely that it is exact. Consider a similar problem of the attainable orderings of $\{a_ia_j\}_{1\leq i\leq j\leq n}$ for n positive reals $a_1>\ldots>a_n>0$ . For the case $n=4$ , the corresponding poset is denoted by the following Hasse diagram of Figure 6 (which is isomorphic to that in Figure 5(a)). But as discussed in Johnston (Reference Johnston2014), the number of attainable linear orderings is 10 and not 12 (the total number of linear orderings consistent with the partial order of Figure 6). In this problem, while the diagonal is involved, the problem is seemingly simpler to analyze than the one we consider as instead of our $\Psi(\cdot,\cdot)$ it involves the simpler binary operator, the product. In the specific case of $d=5$ , all of the 12 possible linear orderings are attainable.

Figure 6. Hasse Diagram – $n=4$ .

In Fiebig et al. (Reference Fiebig, Strokorb and Schlather2017), they investigate the set of all $\{0,1\}$ valued vertices of ${\mathcal{T}}_d$ as ${\mathcal{T}}_d$ being a convex set contains the convex hull of such vertices. In fact, Proposition 21 therein shows by such an analysis that the clique partition polytope is contained in ${\mathcal{T}}_d$ . While the non-convexity of ${\mathcal{T}}'_d$ voids such a rationale, it is nevertheless interesting as a simple measure of the coverage of ${\mathcal{T}}_d$ by ${\mathcal{T}}_d'$ . For this to be meaningful, for this discussion, we extend the class of BB1 copulas to include the lower tail-independent Gumbel copula to accommodate lower tail independence (i.e., tail dependence coefficient of zero). We note that if one of the linking copulae is Gumbel, then the corresponding coordinate is lower tail independent from the rest of the coordinates. This means that the set of all feasible $\{0,1\}$ valued TDMs corresponds to a subset of coordinates being perfectly tail dependent with each other (tail dependence coefficient of 1; comonotonicity copula; $\theta=\infty$ ) and the rest of the coordinates being tail independent with the other $d-1$ coordinates. In particular, the set of feasible $\{0,1\}$ valued TDMs contains $2^d-d$ members out of the dth Bell number of many such vertices of ${\mathcal{T}}_d$ . Using the lower bound for the dth Bell number provided by Berend and Tassa (Reference Berend and Tassa2010), we conclude that that the proportion of $\{0,1\}$ -valued vertices that are feasible is bounded above by

\[ \left( \frac{2 e \log d}{d} \right)^d \to 0, \quad d \to \infty\]

Table 3 presents the exact results for $d\leq 6$ .

Table 3. The number of $\{0,1\}$ -valued and feasible vertices of $\mathcal{T}_d$ , for $3 \leq d \leq 6$ : although the number of $\{0,1\}$ -valued vertices in $\mathcal{T}_d$ (equal to the Bell number) increases super-exponentially with the dimension d, the number of feasible vertices approximately doubles with each increment in dimension.

4. Modeling with one-factor copula

From Theorem 3.1, we know that $\mathcal{T}_d'$ has zero volume for $d \geq 4$ , and moreover, that feasible TDMs are confined to a relatively few sub-regions of ${\mathcal{T}}_d$ . This, in particular, implies that the one-factor copula model’s ability to handle arbitrary tail dependence is quite limited in higher dimensions, similar to the findings in the study by Shyamalkumar and Tao (Reference Shyamalkumar and Tao2022) on t-copula. When an actuarial modeler uses the one-factor copula and relies on actuarial opinion to obtain a target TDM or estimates the TDM using data, there is a potential issue when the copula is unable to accommodate the target or estimated TDM. This is because expert opinion and estimates are inherently imprecise, and the modeler may need to carefully search for a closely approximating TDM within ${\mathcal{T}}_d'$ . In this section, we present an approach for finding such a TDM.

Given an infeasible d-dimensional target TDM $\mathbf{T}_g=(t_{ij})$ , we aim to find the closest feasible TDM in Frobenius norm. To achieve this, we formulate an optimization problem as the following:

(4.1) \begin{equation}\begin{array}{rcl@{}}& \text{minimize} & \mathbf{h}_d(\mathbf{x})\;:\!=\;\frac{1}{2} || \mathbf{\Psi}_d (\mathbf{x})-\mathbf{T}_g||^2=\sum \limits_{1 \leq i<j\leq d} [\Psi(x_{i},x_{j})-t_{i,j}]^2;\\[5pt] \\[5pt] &\text{subject to}& \mathbf{x}=[x_1,x_2,\ldots,x_d] \in \mathbb{R}_+^d.\end{array}\end{equation}

Here, $\mathbf{\Psi}_d(\mathbf{x})$ is the TDM generated by the one-factor copula model with parameters $\mathbf{x}$ , and $\Psi(x_i,x_j)$ denotes the (i,j)-th entry of $\mathbf{\Psi}_d(\mathbf{x})$ . The objective function $\mathbf{h}_d(\mathbf{x})$ measures the distance between the target TDM $\mathbf{T}_g$ and the TDM generated by the one-factor copula model with parameters $\mathbf{x}$ . The optimization problem is subject to the non-negative constraints on $\mathbf{x}$ to ensure the feasibility of the resulting TDM.

Clearly, the definition of $\Psi$ as a definite integral without a closed-form value rules out a closed-form solution to the above optimization problem. Hence, we must rely on numerical optimization methods to solve the optimization problem (4.1). The fmincon function of MATLAB (see MathWorks, 2021) finds local solutions to nonlinear non-convex problems using an interior-point algorithm. It is worth noting that by providing the gradient,

\[\nabla \mathbf{h}_d(\mathbf{x}) = \left(\frac{\partial \mathbf{h}_d(\mathbf{x})}{\partial x_1}, \ldots, \frac{\partial \mathbf{h}_d(\mathbf{x})}{\partial x_d} \right),\]

where

\[\frac{\partial \mathbf{h}_d(\mathbf{x})}{\partial x_i} = \sum_{j \neq i} 2 [\Psi(x_{i},x_{j})-t_{i,j}] \frac{\partial \Psi (x_i,x_j)}{\partial x_i},\]

we can significantly decrease the computational time. However, the built-in trust-region algorithm of the fmincon function performs worse; thus, we do not report its results.

To test the algorithm, we do not have a standard TDM test library available for $d \geq 7$ . Therefore, we utilize two parametric families of TDMs presented in Shyamalkumar and Tao (Reference Shyamalkumar and Tao2020). Firstly, we consider the two-sector matrices:

(4.2) \begin{eqnarray} \begin{bmatrix} 1 &\alpha& \cdots & \alpha & \gamma &\gamma & \cdots & \gamma \\[5pt] \alpha &1& \cdots & \alpha & \gamma &\gamma & \cdots & \gamma \\[5pt] \vdots & \vdots &\ddots &\vdots &\vdots & \vdots &\ddots & \vdots \\[5pt] \alpha &\alpha & \cdots & 1 & \gamma & \gamma & \cdots & \gamma \\[5pt] \gamma & \gamma & \cdots & \gamma & 1 & \beta & \cdots &\beta \\[5pt] \gamma & \gamma & \cdots & \gamma & \beta & 1 & \cdots &\beta \\[5pt] \vdots & \vdots &\ddots &\vdots &\vdots & \vdots &\ddots & \vdots \\[5pt] \gamma & \gamma & \cdots & \gamma & \beta & \beta & \cdots & 1 \end{bmatrix}= \begin{bmatrix} (1-\alpha)\mathbf{I}_{d_1}+\alpha\mathbf{J}_{d_1} & \gamma \mathbf{J}_{d_1 \times d_2} \\[5pt] \gamma \mathbf{J}_{d_2 \times d_1} & (1-\beta)\mathbf{I}_{d_2}+\beta\mathbf{J}_{d_2} \\[5pt] \end{bmatrix}. \end{eqnarray}

where $(\alpha, \beta) \in [0,1]^2$ and $\gamma \in [0, \gamma_u(\alpha,\beta)]$ . The value of $\gamma_u(\alpha,\beta)$ can be computed using a linear programming formulation, as detailed in Shyamalkumar and Tao (Reference Shyamalkumar and Tao2020). In this two-section case, it is natural to model it using a two-factor model. However, in our analysis, we utilize a one-factor copula to model this situation. Secondly, in order to test the algorithm, we consider the two-dependent matrices of the form:

(4.3) \begin{equation} \begin{bmatrix}1 & \alpha & \beta & 0 & \cdots & 0 \\[5pt] \alpha & 1 & \alpha & \ddots & \ddots & \vdots \\[5pt] \beta & \alpha & \ddots & \ddots & \ddots& 0 \\[5pt] 0 & \ddots &\ddots & \ddots & \alpha &\beta \\[5pt] \vdots & \ddots &\ddots & \alpha & 1 & \alpha\\[5pt] 0 & \cdots & 0 &\beta & \alpha &1\end{bmatrix},\end{equation}

where the matrix of the form (4.3) is in $\mathcal{T}_d$ if and only if $\alpha$ and $\beta$ satisfy the following constraints:

(4.4) \begin{eqnarray}\begin{cases}\alpha, \beta \geq 0; \\[5pt] \alpha+4\beta \leq 2; \\[5pt] 2\alpha-\beta \leq 1.\end{cases}\end{eqnarray}

It is important to note that all of the matrices of the form (4.3) are not in ${\mathcal{T}}_d'$ , except when $\alpha=\beta=0$ , by Lemma A.6 of the Appendix.

All the computations in this section were performed on a computer equipped with a single Intel Quad Core i7 2.7GHz processor and 16GB of RAM. To investigate the computational performance of the fmincon function in MATLAB for both the two-sector and two-dependence cases, we conducted experiments with various dimensions. For the two-sector case, we considered dimensions $d=5,10,\ldots,60$ and randomly generated 25 pairs of $(\alpha, \beta) \in [0,1]^2$ for each dimension. We calculated $\gamma_u(\alpha,\beta)$ for each pair and then randomly generated $\gamma \in [0, \gamma_u(\alpha,\beta)]$ . Similarly, for the two-dependence case, we randomly generated 25 pairs $(\alpha, \beta)$ from show convex polytope (4.4) dimensions $5, 10, \ldots, 60$ . For each dimension, we report the running times across all non-linear instances but in Figure 7.

Figure 7. Testing the performance of the interior-point algorithm: test instances were generated randomly by selecting 25 $d \times d$ matrices from two-sector matrices of the form (4.2) and two-dependent matrices of the form (4.3) for $d \in \{5,10, \ldots, 60\}$ .

As shown in the figure, the running times are under four minutes for all instances from both sets of TDMs, for dimensions up to 60. These experimental results show that the interior-point algorithm demonstrates good scaling for our problem. It is important to highlight, however, that the running times exhibit a non-linear, but less than cubic, growth.

5. An application to weather-related insurance

The global increase in both the frequency and severity of natural catastrophes, due to climate change or otherwise, and accompanying increased competition for food resources with the shrinking of arable land have contributed to the importance of crop insurance. Excessive rain and, in extreme cases, flooding or near-drought conditions are key risk events covered by crop insurance, as these can severely disrupt a farming business. From the point of view of an insurer or a reinsurer with a portfolio of crop insurance policies spanning a geographical area, an accurate assessment of the dependence structure of rainfall at different locations, especially in the tails, is a critical input to evaluate the geographical diversification or lack thereof, the latter affecting both pricing and capital considerations. In the following, we show how the methodology above can help develop a one-factor copula model that embodies the tail dependence exhibited in a rainfall dataset. We consider below the daily rainfall data from the European Climate Assessment & Dataset project (Klein Tank et al., Reference Klein Tank, Wijngaard, Können, Böhm, Demarée, Gocheva, Mileta, Pashiardis, Hejkrlik, Kern-Hansen, Heino, Bessemoulin, Müller-Westermeier, Tzanakou, Szalai, Pálsdóttir, Fitzgerald, Rubin, Capaldo, Maugeri, Leitass, Bukantis, Aberfeld, van Engelen, Forland, Mietus, Coelho, Mares, Razuvaev, Nieplova, Cegnar, Antonio López, Dahlström, Moberg, Kirchhofer, Ceylan, Pachaliuk, Alexander and Petrovic2002).Footnote ¹ More precisely, we focus on a recent ten-year data (2011/1/1–2020/12/31) of 33 French cities with 1% of missing value.

A random vector (X,Y) of rainfall in two cites on a given day, as suggested by Shimizu (Reference Shimizu1993), is modeled as described below:

\begin{align*} &\textrm{Pr}\left(X=0,Y=0\right)=\lambda_0; & \textrm{Pr}\left(0< X \leq x,Y=0\right)=\lambda_1 F_{1}(x); \\[5pt] & \textrm{Pr}\left(X = 0,0<Y\leq y\right)=\lambda_2 F_{2}(y); & \textrm{Pr}\left(0< X \leq x,0<Y\leq y\right)=\lambda_3 F(x,y),\end{align*}

where $x,y,\lambda_i>0$ , $i=0,\ldots,3$ satisfy $\sum_{i=0}^3 \lambda_i=1$ , $F_1$ and $F_2$ are univariate continuous distribution functions supported on $(0,\infty)$ , and F is a bivariate continuous joint distribution function supported on the positive quadrant. In the following, as we restrict our interest to the upper tail dependence of rainfall in each pair of cities, we only need to model the joint cumulative distribution function F.

Our approach is to model F by choosing a one-factor copula with parameters $(\tilde{\theta},\tilde{\delta})$ and appropriate marginals $F_X$ and $F_Y$ . Following Serinaldi (Reference Serinaldi2008), we use the L-moments ratio diagrams (see Hosking (Reference Hosking1990)) for the purpose of choosing an appropriate distribution family for modeling the marginals. In Figure 8, the L-skewness versus L-kurtosis graph, we plot the $2\times\binom{33}{2}$ empirical estimates from the marginals of truncated (to $(0,\infty)^2$ ) bivariate distributions along with the curves from some commonly used distribution families. Figure 8 suggests the three-parameter Weibull distribution as a good fit. To estimate the parameters, we use the fact that the Weibull distribution is a reverse Generalized Extreme Value distribution. This allows us to employ the L-moments-based estimators for the parameters of Generalized Extreme Value distributions listed in Table 2 of Hosking (Reference Hosking1990). These estimators are conveniently made available on the R statistical environment by the packages lmom and lmomco (see Hosking, Reference Hosking2019 and Asquith, Reference Asquith2020).

Figure 8. The L-moments ratio diagrams of some common distributions.

Having estimated the marginals as described above, since we seek a model that captures well the tail dependence, we begin by estimating the tail dependence matrix. Among many nonparametric estimators for the tail dependence coefficient, the Reference Capéraà, Fougères and GenestCapéraà-Fougères-Genest estimator of the tail dependence coefficient (see Capéraà et al., Reference Capéraà, Fougères and Genest1997) given by

\[\widehat{\chi}^{\text{CFG}}=2- \prod_{i=1}^n\left( \frac{\log \min (u_{1,i},u_{2,i}) } {\log \max (u_{1,i},u_{2,i}) } \right)^\frac{1}{2n},\]

where n is the sample size, $u_{1,i}=\hat{F}_X(x_{1,i})$ , and $u_{2,i}=\hat{F}_Y(x_{2,i})$ , for $i=1,\ldots,n$ , shows the best performance in, for example Frahm et al. (Reference Frahm, Junker and Schmidt2005). Using this estimator we compute the $33 \times 33$ estimated TDM, denoted by $\widehat{\mathbf{T}}_{33}^{\text{CFG}}$ , which interestingly is a valid TDM.

On the other hand, there are 236 of $5{}456$ , $3 \times 3$ principal submatrices that are determined to be not feasible TDMs by Proposition 3.2. It is noteworthy that 236 represents only about $4.3\%$ of 5456, far smaller a proportion than the 26% relative volume of ${\mathcal{T}}_3'$ reported in Theorem 3.1. This, in particular, implies that $\widehat{\mathbf{T}}_{33}^{\text{CFG}} \in \mathcal{T}_{33} \setminus\mathcal{T}_{33}'$ , suggesting the use of the Frobenius norm closest one-factor feasible TDM, say $\widetilde{\mathbf{T}}_{33}$ , to $\widehat{\mathbf{T}}_{33}^{\text{CFG}}$ for estimating the parameters of a one-factor copula. As the upper tail-dependence coefficient only determines $\theta\delta$ , for our illustrative purposes we fix $\delta = 1.2$ , but in practice, one can use constrained maximum likelihood or any other suitable statistical paradigms.

To illustrate the goodness of fit resulting from the above inference procedure, we consider two sets of three cities. The first set consisting of Bourges, Nice, and Saint-Girons is chosen as its corresponding principal submatrix of $\widehat{\mathbf{T}}_{33}^{\text{CFG}}$ is not in $\mathcal{T}_{3}'$ ; this is so as according to Proposition 3.2, infeasibility is implied by

\[ \Psi \left(\Psi_1 ^{-1}(0.0820),\Psi_1 ^{-1}(0.2254) \right)= \Psi \left(0.2772,0.4652 \right) =0.0325 >0.0003. \]

This matrix, its counterpart in the feasible TDM $\widetilde{\mathbf{T}}_{33}$ , and the corresponding one-factor copula parameters are given by

\[\begin{bmatrix} 1 & 0.0820 & 0.2254\\[5pt] 0.0820 & 1 & 0.0003\\[5pt] 0.2254 & 0.0003 & 1\end{bmatrix},\;\begin{bmatrix}1 & 0.1011 & 0.2459 \\[5pt] 0.1011 & 1 & 0.0776 \\[5pt] 0.2459 & 0.0776 &1\end{bmatrix},\hbox{ and } \begin{bmatrix} 1.0032\\[5pt] 0.3502\\[5pt] 0.6662\end{bmatrix},\]

respectively. The above being quite similar suggests a good fit in the upper tails.

In Figure 9, the contour plots of the bivariate marginals from the fitted one-factor copula, with Weibull marginals, for these three cities being quite alike the contours of the kernel density estimate suggests a good overall fit. For this plot, we used the function sim1fact in the R package CopulaModel, see Joe (Reference Joe2014).

Figure 9. Contours of the fitted kernel and one-factor-copula-based distribution (dotted lines).

The second set of cities consisting of Bourges, Lyon, and Strasbourg was chosen as its corresponding principal submatrix of $\widehat{\mathbf{T}}_{33}^{\text{CFG}}$ is in $\mathcal{T}_{3}'$ ; this is so as according to Proposition 3.2, feasibility is implied by

\[ \Psi \left(\Psi_1 ^{-1}(0.2620),\Psi_1 ^{-1}(0.2842) \right)= \Psi \left(0.5174,0.5510 \right) =0.1168 < 0.2302. \]

This matrix, its counterpart in $\widetilde{\mathbf{T}}_{33}$ , and the corresponding one-factor copula parameters are given by

\[\begin{bmatrix}1 & 0.2842 & 0.2620 \\[5pt] 0.2842 & 1 & 0.2302 \\[5pt] 0.2620 & 0.2302 &1\end{bmatrix},\;\begin{bmatrix}1 & 0.2714 & 0.2772 \\[5pt] 0.2714 & 1 & 0.2267 \\[5pt] 0.2772 & 0.2267 &1\end{bmatrix}, \hbox{ and } \begin{bmatrix} 1.0032\\[5pt] 0.7444\\[5pt] 0.7636\end{bmatrix},\]

respectively. Note that these matrices are nearly identical. Also, the contour plots of the bivariate marginals from the fitted one-factor copula for these three cities are similar to the contours of the kernel density estimate, as shown in Figure 10, suggesting a good overall fit. Relatively speaking, notice that we see a better alignment of contours in Figure 10 than in Figure 9, agreeing with the observations on the empirical TDM.

Figure 10. Contours of the fitted kernel and one-factor-copula-based distribution (dotted lines).

Among the 40,920, $4 \times 4$ principal submatrices, as expected by Theorem 3.1, none are feasible as their distance to the nearest feasible matrix using the optimization problem of (4.1) is positive. Propositions 3.3 and 3.4, by partitioning these matrices into cells depicted in Figure A1, give a qualitative description of proximity to feasibility. In particular, 5691 are located in cells 1 or 7. Additionally, 11,117 matrices are situated in cells adjacent to cells 1 or 7, namely cells 2, 3, 8, and 13. To be precise, what we mean by the qualitative statement is the stochastic ordering depicted in Figure 11, which plots the kernel density estimates of the log-squared distance of each member to feasibility categorized by their membership in the cells. These distances were computed using the optimization problem of (4.1).

Figure 11. Kernel density estimate plots for the three groups.

Consider the following set of four cities: Strasbourg, Nantes, Troyes, and Caen. Their principal submatrix within $\widehat{\mathbf{T}}_{33}^{\text{CFG}}$ is not in the cells 1 or 7 of Table A2. This matrix, its counterpart in $\widetilde{\mathbf{T}}_{33}$ , and the corresponding one-factor copula parameters are given by

\[\begin{bmatrix} 1 & 0.2553 & 0.3106 & 0.2761\\[5pt] 0.2553 & 1 & 0.2691 & 0.3451\\[5pt] 0.3106 & 0.2691 & 1 & 0.3587\\[5pt] 0.2761 & 0.3451 &0.3587 &1\end{bmatrix},\;\begin{bmatrix} 1 & 0.2569& 0.2801 & 0.2893\\[5pt] 0.2569 & 1 & 0.3124 & 0.3230\\[5pt] 0.2801 & 0.3124 & 1 & 0.3538\\[5pt] 0.2893 &0.3230 & 0.3538 &1\end{bmatrix}, \hbox{ and } \begin{bmatrix} 0.7636\\[5pt] 0.8846\\[5pt] 1.0225\\[5pt] 1.0881\end{bmatrix}.\]

Proposition 3.4 indicates that $\mathcal{T}_4'$ can be divided into $4!=24$ distinct subsets, each reflecting a permutation of the set $\{1,2,3,4\}$ , aligning with the sequence of the $\eta_i$ values. Furthermore, according to Proposition 3.3, these subsets can be subdivided based on permutations of $\mathcal{I}_4$ , arranged according to the tail dependence coefficients. Due to symmetry, this results in $6!/4! = 30$ such subdivisions. Notably, among these, only two subsets are nonempty. Located in cell 9 of Table A2 and Figure A1, the matrix’s projection falls within cell 1, preserving the ordering of the underlying $\eta_i$ ’s. Given the proximity of the TDM projection, this is not surprising.

6. Broader implications of our analysis

To begin with, we note that the tail dependence coefficients of the following linking copulas share the same functional form as that of the BB1 family: Clayton copula (see, section 7.1 of McNeil et al., Reference McNeil, Frey and Embrechts2015 or Family B4 in section 5.1 of Joe, Reference Joe1997), Galambos survival copula (see, section 4.9 of Joe, Reference Joe2014), BB4 survival copula (see, section 5.2 of Joe, Reference Joe1997), and BB7 copula (see, Table A1 in the Appendix). Hence, our results in Section 3 carry over to the one-factor model with these linking copulas as well.

Suppose that the tail dependence coefficient is defined by $\Psi\;:\; \Theta^2 \to [0,1]$ , with the parameter space $\Theta$ being an interval of $\mathbb{R}$ . In other words, we suppose that the i-th coordinate is associated with a parameter $\theta_i$ (possibly with parametrization, see FDG copulas below) such that the tail dependence coefficient between the i-th and the j-th coordinate equals $\Psi(\theta_i,\theta_j)$ . Further, if $\Psi$ is a symmetric, strictly monotone (not necessarily increasing) function with continuous derivatives, then the results outlined in Section 3 materially continue to hold. Only the computation of the volume and the non-convexity of $\mathcal{O}'_{\!\!3}$ referred to in Remark 3.1 use the specificity of the BB1 copula.

Examples of such linking copulas include the Gumbel survival copula (see, Section 7.1 of McNeil et al., Reference McNeil, Frey and Embrechts2015), B5 survival copula (see, Section 5.1 of Joe, Reference Joe1997), BB1 survival copula, and BB3 survival copula (see, Section 5.2 of Joe, Reference Joe1997). Interestingly, for all of these examples, the tail dependence coefficient between two coordinates has the form,

\begin{align*}\begin{split} \Psi (x,y)&\;:\!=\;\int_0 ^ \infty \left[1- \left(1+ w^{-x} \right)^{\frac{1}{x}-1} \right] \left[ 1- \left(1+ w^{ -y} \right)^{\frac{1}{y}-1} \right] dw,\\[5pt] \hbox{with } \Psi (x,\infty)&\;:\!=\;\Psi(\infty,x)=1-\int_0^1 (1+w^{-x})^{\frac{1}{x}-1} dw=2-2^{\frac{1}{x}}, \\[5pt] \hbox{and } \Psi(\infty,\infty)&\;:\!=\;1, \qquad 1 \leq x,y <\infty. \end{split}\end{align*}

Note that the above $\Psi$ is a symmetric strictly increasing function on $[1,\infty]^2$ with continuous partial derivatives in the interior.

FDG copula family – one-factor copula with Durante Generators – is a one-factor copula model with the Durante class of copulas, a tractable nonparametric class, as linking copulas. See Mazo et al. (Reference Mazo, Girard and Forbes2016) for the FDG copula and Durante (Reference Durante2006) for the Durante class.

The Durante class of copulas is given by

\[C(u,v) \;:\!=\; (u\wedge v) \cdot f(u\vee v), \quad u,v\in[0,1]^2,\]

where $f\;:\;[0,1] \to [0,1]$ , the generator, is a differentiable increasing function satisfying $f(1)=1$ with $u\mapsto f(u)/u$ a decreasing function. Note that the copula represents an exchangeable distribution, and by considering the conditional distribution of X given $Y\leq v$ , it is easy to see the need for the restrictions on f with differentiability added for the existence of the density. The lower and upper tail dependence coefficients are f(0) and $1-f'(1)$ , respectively.

It is interesting to note that the bivariate margins of an FDG copula are themselves members of the Durante class. Hence, if $f_1$ and $f_2$ are the generators of the first two coordinates, then their bivariate margin is a Durante class copula with generator

\[u\mapsto f_1(u)\cdot f_2(u) + u\int_u^1 f_1'(v)f_2'(v) \textrm{d}v.\]

In particular, $f_1(0)\cdot f_2(0)$ and $(1-f'_1(1))(1-f'_2(1))$ are its lower and upper tail dependence coefficients, respectively. This implies that the ordering of the tail dependence coefficients of the linking copula imposes a partial ordering on the tail dependence coefficients of the FDG copula. Some parametric examples of generators are the Cuadras-Augé generator $u\mapsto u^{1-\theta}$ for $\theta\in[0,1]$ , Fréchet generator $u\mapsto (1-\theta)u+\theta$ , for $\theta \in[0,1]$ and the Durante-exponential generators

$$ u \mapsto \exp\left(\frac{u^{\theta}-1}{\theta}\right), \qquad \theta>0.$$

Finally, since for any copula C, $C^\ast(u_1,u_2)\;:\!=\;C(1-u_1,1-u_2)$ also forms a copula, and the lower tail dependence coefficient with respect to C corresponds to the upper tail dependence coefficient with respect to $C^\ast$ , and vice versa, the applicability of our results is invariant to the focus being on the left or the right tail. Moreover, for copula families like the FDG copula, the results apply to both the tails of the copula.

7. Conclusion

In statistical modeling with reasonable complexity, it is common that explicit modeling choices sweep under the carpet implicit model assumptions, some of which may be unjustified. In this study, we investigate if this happens in the context of modeling high-dimensional data that exhibits nonlinear tail-dependence when we use a one-factor copula model with a parametric family for the linking copulas. Apart from deriving basic tail dependence properties for this mode, we precisely describe the implicit ordering of the tail dependence coefficients, an implementation of a solution to the formulated optimization problem to help model data from a tail dependence perspective and conclude with an illustrative application.

We believe this paper’s qualitative implications are broad; as a next step, an extension to the multifactor setting will be interesting as we continue to retain the linear-in-dimension parsimonious parameterization of the one-factor copula models.