On a Measure of Tail Asymmetry for the Bivariate Skew-Normal Copula

1. Introduction

Skewness in data is an important feature in various applications. As a typical example, dependence among stock returns is known to be asymmetric in bearish and bullish markets [1]. To capture such asymmetry, skewed distributions have been extensively studied in the literature. A popular family of skewed distributions is the skew elliptical distribution [2], which includes, for example, the skew-normal distribution [3] and the skew-t distribution [4]. Various measures of asymmetry are also proposed, for example, in [5,6,7,8,9] to quantify certain asymmetric feature of an underlying distribution, typically an asymmetry between upper and lower tails.

For a more formal description of the subject of this paper, let $\mathbf{X}=(X_1,X_2)$ be an ${\mathbb{R}}^2$-valued random vector on a fixed atomless probability space. Denote by H the cumulative distribution functions (cdf) of $\mathbf{X}$ with marginal distributions $F_1$ and $F_2$, respectively. For a fixed threshold $u\in (0,1/2]$, we focus on the following measure of tail asymmetry (in lower and upper tails) [9]:

$$\begin{array}{c}\hfill {\nu }_H\left(u\right)=log\left(\frac{\mathbb{P}(X_1\ge F_1^{-1}(1-u),X_2\ge F_2^{-1}(1-u))}{\mathbb{P}(X_1\le F_1^{-1}\left(u\right),X_2\le F_2^{-1}\left(u\right))}\right).\end{array}$$

(1)

We are also interested in the asymptotic behavior of ${\nu }_H\left(u\right)$ as $u\downarrow 0$. Throughout the paper, we assume that $F_1$ and $F_2$ are continuous so that $(X_1,X_2)$ has the unique copulaC, which is the cdf of $(U_1,U_2)=(F_1\left(X_1\right),F_2\left(X_2\right))$. Then ${\nu }_H$ is a function of the copula given by

$$\begin{array}{cc}\hfill {\nu }_H\left(u\right)& =log\left(\frac{\mathbb{P}(U_1\ge 1-u,U_2\ge 1-u)}{\mathbb{P}(U_1\le u,U_2\le u)}\right)\hfill \\ \hfill & =log\left(\frac{\overline{C}(1-u,1-u)}{C(u,u)}\right)=:{\nu }_C\left(u\right),\hfill \end{array}$$

where $\overline{C}$ is the copula of $(1-U_1,1-U_2)$.

This paper concerns the measure of tail asymmetry (1) and its asymptotic behavior of the skew-normal distribution. A d-dimensional random vector $\mathbf{Y}=(Y_1,\cdots ,Y_n)$ is said to follow the skew-normal distribution, denoted by $\mathbf{Y}\sim {SN}_d(\delta ,\mathsf{\Psi })$, if it admits the stochastic representation

$$\begin{array}{c}\hfill Y_j={\delta }_j\left|Z_0\right|+\sqrt{1-{\delta }_j^2}{Z}_j,{\delta }_j\in (-1,1),\mathsf{\Psi }\in {\mathcal{P}}_d,\end{array}$$

(2)

where $Z_0\sim N(0,1)$, $\mathbf{Z}=(Z_1,\cdots ,Z_d)\sim N_d(\mathbf{0},\mathsf{\Psi })$ is independent of $Z_0$ and ${\mathcal{P}}_d$ is a set of all d-dimensional correlation matrices. The parameter $\delta =({\delta }_1,\cdots ,{\delta }_d)\in {(-1,1)}^d$ is called the skewness parameter and $\mathsf{\Psi }\in {\mathcal{P}}_d$ is called the correlation matrix. The skew-normal copula, denoted by $C_{SN}(·;\delta ,\mathsf{\Psi })$, is defined as the copula of $\mathbf{Y}\sim {SN}_d(\delta ,\mathsf{\Psi })$.

The contribution of this paper is twofold. First, we derive an asymptotic formula of the measure of tail asymmetry (1) in terms of the upper and lower tail orders [10]. This formula enables us to describe the asymptotic behavior of this measure for the skew-normal copula. Various approaches are also introduced to evaluate the measure (1) for a finite threshold $u\in (0,1/2]$. Our second contribution is to numerically demonstrate that, when the correlation or skewness parameters are around at the boundary values, some asymptotic formulas of the skew-normal copula proposed in the literature are not suitable to compute the measure of tail asymmetry with practically extremal thresholds, such as $u=0.01$. This finding supports the use of an exact evaluation of the measure (1) even for an extremely small threshold $u\in (0,1/2]$ instead of some asymptotic tail approximations proposed in the literature.

Apart from the measure (1), some other measures of tail asymmetry have been proposed in the literature. Many of these existing measures are based on differences between certain measures of upper and lower tails (see, e.g., [5,6,7,8]). Such difference-based measures are not appropriate to quantify tail asymmetry of the skew-normal copula since values of these measures tend to be small even for large values of $\parallel \delta \parallel$. On the other hand, the measure (1) returns reasonable values for such values of $\delta$ because this measure is based on the ratio between upper and lower tail probabilities. To the best of our knowledge, the measure (1) is the only ratio-based measure of upper and lower tail probabilities, which seems appropriate to measure the tail asymmetry of the skew-normal copula.

The organization of this paper is as follows. In Section 2, we review the concept of tail order and skew-normal copulas. In Section 3, we derive a formula of the measure of tail asymmetry in terms of the upper and lower tail orders. We also explore the measure and its asymptotic behavior for the skew-normal copula. Numerical experiments are provided in Section 4, where we reveal some situations when some asymptotic formulas of the skew-normal copula proposed in the literature are not appropriate to use . Section 5 concludes this paper. Detailed calculations and all proofs are deferred to Appendix A and Appendix B, respectively.

2. Preliminaries

We begin with introducing some concepts and notations. Two functions $f,g:\mathbb{R}\to \mathbb{R}$ are called asymptotically equivalent at $a\in \overline{\mathbb{R}}=\mathbb{R}\cup \{\pm \infty \}$, denoted by $f\left(x\right)\sim g\left(x\right)$, $x\to a$, if ${lim}_{x\to a}f\left(x\right)/g\left(x\right)=1$. A function $f:\mathbb{R}\to \mathbb{R}$ is called slowly varying at $a\in [0,\infty ]$ if ${lim}_{x\to a}f\left(tx\right)/f\left(x\right)=1$ for any $t\in (0,\infty )$. The set of all slowly varying functions at a is denoted by ${SV}_a$. Throughout the paper, all vectors in the form $(x_1,\cdots ,x_n)$, $n\in \mathbb{N}$, are understood as column vectors.

2.1. Tail order and tail order parameter

According to [10], a d-dimensional copula C is said to have the lower tail order${\kappa }_L\left(C\right)\ge 1$ if

$$\begin{array}{c}\hfill C(u,\cdots ,u)\sim u^{{\kappa }_L\left(C\right)}{\ell }_L\left(u\right),u\downarrow 0,\end{array}$$

(3)

where ${\ell }_L\in {SV}_0$. If, in addition, the limit ${lim}_{u\downarrow 0}{\ell }_L\left(u\right)={\ell }_L^{*}\left(C\right)$ exists, then C is said to have the lower tail order parameter${\ell }_L^{*}\left(C\right)\in [0,\infty ]$. The copula C is called (lower) tail dependent when ${\kappa }_L\left(C\right)=1$. In this case, ${\ell }_L^{*}\left(C\right)$ is known as the (lower) tail dependence coefficient (TDC). The case ${\kappa }_L\left(C\right)=d$ is referred to as the tail independence. When $1<{\kappa }_L\left(C\right)<d$, C is said to have intermediate tail dependence. As such the model (3) can capture weaker tail dependence that the TDC cannot. Similarly to the lower case, C is said to have the upper tail order${\kappa }_U\left(C\right)\ge 1$ and the upper tail order parameter${\ell }_U^{*}\left(C\right)\in [0,\infty ]$ if

$$\begin{array}{c}\hfill \overline{C}(1-u,\cdots ,1-u)\sim u^{{\kappa }_U\left(C\right)}{\ell }_U\left(u\right),u\downarrow 0,\end{array}$$

(4)

where ${\ell }_U\in {SV}_0$ is such that ${lim}_{u\downarrow 0}{\ell }_U\left(u\right)={\ell }_U^{*}\left(C\right)$.

2.2. The skew-normal copula

Let $\mathbf{Y}\sim {SN}_d(\delta ,\mathsf{\Psi })$ follow a d-dimensional skew-normal distribution defined via the stochastic representation (2). According to [3], the joint probability density functions (pdf) of $\mathbf{Y}$ is given by

$$\begin{array}{cc}\hfill f_{SN}(\mathbf{y};\alpha ,\mathsf{\Omega })& =2{\varphi }_d(\mathbf{y};\mathsf{\Omega })\mathsf{\Phi }\left({\alpha }^{\top }\mathbf{y}\right),\mathbf{y}\in {\mathbb{R}}^d,\hfill \end{array}$$

where ${\varphi }_d(·;\mathsf{\Omega })$ is the pdf of $N_d(\mathbf{0},\mathsf{\Omega })$ and the parameters $\mathsf{\Omega }\in {\mathcal{P}}^d$ and $\alpha \in {\mathbb{R}}^d$ are specified via

$$\begin{array}{cc}\hfill \mathsf{\Omega }& =\mathsf{\Delta }(\mathsf{\Psi }+\zeta {\zeta }^{\top })\mathsf{\Delta },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \alpha & =\frac{{\mathsf{\Omega }}^{-1}\delta }{\sqrt{1-{\delta }^{\top }{\mathsf{\Omega }}^{-1}\delta }}=\frac{{\mathsf{\Delta }}^{-1}{\mathsf{\Psi }}^{-1}\zeta }{\sqrt{1+{\zeta }^{\top }{\mathsf{\Psi }}^{-1}\zeta }},\hfill \end{array}$$

(6)

where

$$\begin{array}{cc}\hfill \mathsf{\Delta }& =diag\left(\sqrt{1-{\delta }_1^2},\cdots ,\sqrt{1-{\delta }_d^2}\right),\hfill \\ \hfill \zeta & =({\zeta }_1,\cdots ,{\zeta }_d),{\zeta }_j=\frac{{\delta }_j}{\sqrt{1-{\delta }_j^2}},j\in \{1,\cdots ,d\}.\hfill \end{array}$$

The marginal pdf of $Y_j$, $j\in \{1,\cdots ,d\}$, is given by

$$\begin{array}{cc}\hfill f_{SN}(y_j;{\delta }_j)& =2\varphi \left(y_j\right)\mathsf{\Phi }\left({\zeta }_j{y}_j\right),y_j\in \mathbb{R},\hfill \end{array}$$

where $\varphi$ and $\mathsf{\Phi }$ are pdf and cdf of $N(0,1)$, respectively. Note that $Y_j\sim {SN}_1\left({\delta }_j\right)$, $j\in \{1,\cdots ,d\}$, where ${SN}_1\left({\delta }_j\right)={SN}_1({\delta }_j,1)$. Moreover, it follows from (2) that

$$\begin{array}{c}\hfill -\mathbf{Y}\sim SN(-\delta ,\mathsf{\Psi }).\end{array}$$

(7)

Skewness of the skew-normal copula is illustrated in Figure 1, where the contour plot of the (symmetric) normal distribution is compared with that of the skew-normal copula with its marginal distributions transformed into the standard normal distributions.

The skew-normal distribution can be written in terms of the conditional distribution of the normal distribution. More precisely, it is shown in ([3], Section 2.2) that

$$\begin{array}{c}\hfill (X_1,\cdots ,X_d)\mid \{X_0>0\}\sim SN(\delta ,\mathsf{\Psi }),\end{array}$$

(8)

where $(X_0,X_1,\cdots ,X_d)\sim N_{d+1}({\mathbf{0}}_{d+1},{\mathsf{\Omega }}^{*}\left(\delta \right))$ with the extended correlation matrix

$$\begin{array}{c}\hfill {\mathsf{\Omega }}^{*}\left(\delta \right)=\left(\begin{array}{cc}1& {\delta }^{\top }\\ \delta & \mathsf{\Omega }\end{array}\right)\in {\mathcal{P}}_{d+1}.\end{array}$$

(9)

This representation allows us to write the cdfs of ${SN}_d(\delta ,\mathsf{\Psi })$ and its copula as follows.

Lemma 1. 

Let $F_{SN}(·;\delta ,\mathsf{\Psi })$ be the cdf of $\mathbf{Y}\sim {SN}_d(\delta ,\mathsf{\Psi })$ with marginal cdfs $F_{SN}(·;{\delta }_j)$, $j\in \{1,\cdots ,d\}$. Then

$$\begin{array}{c}\hfill F_{SN}(\mathbf{y};\delta ,\mathsf{\Psi })=2{\mathsf{\Phi }}_{d+1}((0,\mathbf{y});{\mathsf{\Omega }}^{*}(-\delta )),\end{array}$$

(10)

where ${\mathsf{\Phi }}_{d+1}(·;{\mathsf{\Omega }}^{*}(-\delta ))$ is the cdf of $N_{d+1}({\mathbf{0}}_{d+1},{\mathsf{\Omega }}^{*}(-\delta ))$. Therefore, the cdf of the skew-normal copula $C_{SN}(·;\delta ,\mathsf{\Psi })$ can be written by

$$\begin{array}{c}\hfill C_{SN}(\mathbf{u};\delta ,\mathsf{\Psi })=2{\mathsf{\Phi }}_{d+1}((0,F_{SN}^{-1}(u_1;{\delta }_1),\cdots ,F_{SN}^{-1}(u_d;{\delta }_d));{\mathsf{\Omega }}^{*}(-\delta )).\end{array}$$

(11)

3. Tail asymmetry of the skew-normal copula

In this section we explore tail asymmetry of the skew-normal copula via the measure (1) and its asymptotic behavior. We will show in Proposition 1 that the measure (1), after properly scaled, quantifies the difference between the upper and lower tail orders when they differ. Moreover, when the upper and lower tail orders coincide, the measure (1) quantifies the difference between the upper and lower tail order parameters. These results support the use of the measure (1) for quantifying tail asymmetry of the skew-normal copula since, to our knowledge, difference-based measures of asymmetry proposed in the literature do not measure tail asymmetry properly when the upper and lower tail indices are not equal.

3.1. Measure of tail asymmetry and tail order

This section explores the relationship between the measure of tail asymmetry (1) and the tail order. To this end, suppose that a bivariate copula C satisfies (3) and (4).

The next proposition states that the measure of tail asymmetry (1) can be asymptotically represented in terms of the difference between the upper and lower tail indices.

Proposition 1 

(Measure of tail asymmetry and tail order). Let C be a bivariate copula with lower and upper tail orders ${\kappa }_L\left(C\right)$ and ${\kappa }_U\left(C\right)$, respectively. If ${\kappa }_U\left(C\right)\ne {\kappa }_L\left(C\right)$, then

$$\begin{array}{c}\hfill {\nu }_C\left(u\right)\sim \{{\kappa }_U\left(C\right)-{\kappa }_L\left(C\right)\}logu,u\downarrow 0.\end{array}$$

(12)

If ${\kappa }_U\left(C\right)={\kappa }_L\left(C\right)$ and C admits the upper and lower tail order parameters ${\ell }_U^{*}\left(C\right),{\ell }_L^{*}\left(C\right)\in (0,\infty )$, then

$$\begin{array}{c}\hfill {\nu }_C\left(u\right)\sim log\left(\frac{{\ell }_U^{*}\left(C\right)}{{\ell }_L^{*}\left(C\right)}\right).\end{array}$$

(13)

Remark 1 

(Relationship with TDCs). Equation (13) implies that the limit of ${\nu }_C\left(u\right)$ as $u\downarrow 0$ is obtainable from upper and lower TDCs. Indeed, if ${\kappa }_U\left(C\right)={\kappa }_L\left(C\right)=1$, ${\ell }_U^{*}\left(C\right)$ and ${\ell }_L^{*}\left(C\right)$ correspond to upper and lower TDCs, respectively. Then ${lim}_{u\downarrow 0}{\nu }_C\left(u\right)$ is a straightforward calculation from (13). This result can be applied to evaluate the limit of ${\nu }_C\left(u\right)$ of, for example, the skew-t distribution, whose upper and lower TDCs are available; see [11].

3.2. Measure of tail asymmetry of the skew-normal copula

From this section we focus on the bivariate case $d=2$. Let $\tilde{\rho }\in (-1,1)$ be the off-diagonal entry of $\mathsf{\Psi }$ and $\rho \in (-1,1)$ be that of $\mathsf{\Omega }$. We denote, for example, the bivariate case of $C_{\mathrm{SN}}(·;\delta ,\mathsf{\Psi })$ by $C_{\mathrm{SN}}(·;\delta ,\tilde{\rho })$ for notational simplicity. Note that $\tilde{\rho }$ is the partial correlation of $Y_1$ and $Y_2$ given $Z_0$, where $Z_0,Y_1$ and Y are those used in (2). By calculation, it holds that

$$\begin{array}{c}\hfill \rho =\tilde{\rho }\sqrt{(1-{\delta }_1^2)(1-{\delta }_2^2)}+{\delta }_1{\delta }_2.\end{array}$$

(14)

By selecting $({\delta }_1,{\delta }_2)\in {(-1,1)}^2$ and $\tilde{\rho }\in (-1,1)$ independently, the range of the parameter $\rho$ implied by (14) is given by

$$\begin{array}{c}\hfill {\delta }_1{\delta }_2-\sqrt{(1-{\delta }_1^2)(1-{\delta }_2^2)}<\rho <{\delta }_1{\delta }_2+\sqrt{(1-{\delta }_1^2)(1-{\delta }_2^2)}.\end{array}$$

(15)

Moreover, in the bivariate case, the parameter $\alpha$ is given by

$$\begin{array}{c}\hfill \alpha =\frac{1}{\sqrt{(1-{\rho }^2)(1-{\tilde{\rho }}^2)(1-{\delta }_1^2)(1-{\delta }_2^2)}}\left(\begin{array}{c}{\delta }_1-\rho {\delta }_2\\ {\delta }_2-\rho {\delta }_1\end{array}\right).\end{array}$$

(16)

The reader is referred to Appendix A.1 for detailed derivations of (14) and (16).

We first consider the case of a finite threshold $u\in (0,1/2]$. In this case, the measure of tail asymmetry (1) of the skew-normal copula can be evaluated by the following proposition. Note that we denote by ${\nu }_{SN}(u;\delta ,\tilde{\rho })$ the measure (1) of the skew-normal copula $C_{SN}(·;\delta ,\tilde{\rho })$.

Proposition 2 

(Measure of tail asymmetry of the skew-normal copula). Let $C_{SN}(·;\delta ,\tilde{\rho })$ be the skew-normal copula with $\delta \in {(-1,1)}^2$ and $\tilde{\rho }\in (-1,1)$. Then its measure of tail asymmetry (1) for a finite threshold $u\in (0,1/2]$ is given by

$$\begin{array}{cc}\hfill {\nu }_{SN}(u;\delta ,\tilde{\rho })& =log\left(\frac{C_{SN}(u,u;-\delta ,\tilde{\rho })}{C_{SN}(u,u;\delta ,\tilde{\rho })}\right)\hfill \\ \hfill & =log\left(\frac{{\mathsf{\Phi }}_3\left((0,F_{SN}^{-1}(u;-{\delta }_1),F_{SN}^{-1}(u;-{\delta }_2));{\mathsf{\Omega }}^{*}\left(\delta \right)\right)}{{\mathsf{\Phi }}_3\left((0,F_{SN}^{-1}(u;{\delta }_1),F_{SN}^{-1}(u;{\delta }_2));{\mathsf{\Omega }}^{*}(-\delta )\right)}\right),\hfill \end{array}$$

(17)

where

$$\begin{array}{c}\hfill {\mathsf{\Omega }}^{*}\left(\delta \right)=\left(\begin{array}{ccc}1& {\delta }_1& {\delta }_2\\ {\delta }_1& 1& \tilde{\rho }\sqrt{1-{\delta }_1^2}\sqrt{1-{\delta }_2^2}+{\delta }_1{\delta }_2\\ {\delta }_2& \tilde{\rho }\sqrt{1-{\delta }_1^2}\sqrt{1-{\delta }_2^2}+{\delta }_1{\delta }_2& 1\end{array}\right).\end{array}$$

The formula (17) enables us to numerically compute ${\nu }_{SN}(u;\delta ,\tilde{\rho })$ for a finite threshold $u\in (0,1/2]$.

For an illustrative example, Figure 2 provides curves of ${\nu }_{SN}(u;\delta ,\tilde{\rho })$, $u=0.01$, for the skew-normal copula with different parameters. The function ${\mathsf{\Phi }}_3$ in (17) is evaluated by pmvnorm(algorithm = TVPACK) [12] in the R package mvtnorm. We observe that the measure ${\nu }_{SN}(u;\delta ,\tilde{\rho })$ is symmetric in $\delta$ with respect to $\delta =0$, and is monotonically changing in $\tilde{\rho }$.

We next consider the asymptotic behavior of $\nu \left(u\right)$ for an extremely large and small thresholds. Summarizing the existing results in the literature, we have that

$$\begin{array}{c}\hfill {\kappa }_L(C_{SN};\delta ,\tilde{\rho })=\left\{\begin{array}{cc}\frac{2}{1+\tilde{\rho }},\hfill & \mathrm{if}{\delta }_1,{\delta }_2\ge 0,\hfill \\ \frac{2}{1+\rho },\hfill & \mathrm{if}{\delta }_1,{\delta }_2\le 0;\hfill \end{array}\right.\end{array}$$

(18)

see Appendix A.2 for detailed calculations. Together wtih ${\kappa }_U(C_{SN};\delta ,\tilde{\rho })={\kappa }_L(C_{SN};-\delta ,\tilde{\rho })$, we obtain the following result.

Proposition 3 

(Asymptotic behavior of the measure of tail asymmetry of the skew-normal copula). Let $C_{SN}(·;\delta ,\tilde{\rho })$ be the skew-normal copula with $\delta =({\delta }_1,{\delta }_2)\in {(-1,1)}^2$ and $\tilde{\rho }\in (-1,1)$. Suppose that ${\delta }_1$ and ${\delta }_2$ have the same sign, which includes the case when at least one of them is zero. Then the measure of tail asymmetry (1) of $C_{SN}(·;\delta ,\tilde{\rho })$ satisfies

$$\begin{array}{c}\hfill \underset{u\downarrow 0}{lim}\frac{{\nu }_{SN}(u;\delta ,\tilde{\rho })}{logu}=sign({\delta }_1,{\delta }_2)\left(\frac{2}{1+\rho }-\frac{2}{1+\tilde{\rho }}\right),\end{array}$$

(19)

where

$$\begin{array}{c}\hfill sign({\delta }_1,{\delta }_2)=\left\{\begin{array}{cc}1,\hfill & if{\delta }_1\ge 0and{\delta }_2\ge 0,\hfill \\ -1,\hfill & if{\delta }_1\le 0and{\delta }_2\le 0.\hfill \end{array}\right.\end{array}$$

Remark 2 

(${\delta }_1$ and ${\delta }_2$ with opposite signs). It is assumed in Proposition 3 that ${\delta }_1$ and ${\delta }_2$ have the same sign. However, the upper and lower tail orders of the bivariate skew-normal copula are investigated in [13] for more general ${\delta }_1$ and ${\delta }_2$. Based on their results, analytical expression of the limit ${lim}_{u\downarrow 0}{\nu }_{SN}(u;\delta ,\tilde{\rho })/logu$ can be derived even for ${\delta }_1$ and ${\delta }_2$ with opposite signs although the expression may not be as simple as (19).

Note that ${\nu }_{SN}(u;\delta ,\tilde{\rho })=0$ for every $u\in (0,1/2]$ if $\delta =0$. The asymptotic behavior of ${\nu }_{SN}(u;\delta ,\tilde{\rho })$ as $u\downarrow 0$ is illustrated in Figure 3, where ${\nu }_{SN}(u;\delta ,\tilde{\rho })$ is evaluated by (17) with the algorithm TVPACK. It is observed that the curves of ${\nu }_{SN}(u;\delta ,\tilde{\rho })$ against $-log\left(u\right)$ are asymptotically linear.

Remark 3 

(Measure of tail asymmetry and TDCs). The measure of tail asymmetry (1) of the skew-normal copula can also be written by

$$\begin{array}{c}\hfill {\nu }_{SN}(u;\delta ,\tilde{\rho })=log\left(\frac{{\lambda }_L(u;-\delta ,\tilde{\rho })}{{\lambda }_L(u;\delta ,\tilde{\rho })}\right),\end{array}$$

(20)

where

$$\begin{array}{c}\hfill {\lambda }_L(u;\delta ,\tilde{\rho })=\frac{C_{SN}(u,u;\delta ,\tilde{\rho })}{u}.\end{array}$$

(21)

The value ${\lambda }_L(u;\delta ,\tilde{\rho })$ for a finite $u\in (0,1/2]$ can be computed by (17) with the algorithmTVPACK. Figure 4 shows the curves of $log{\lambda }_L(0.01;\delta ,\tilde{\rho })$ against δ for various correlation parameters. It is observed that ${\lambda }_L(0.01;\delta ,\tilde{\rho })$ increases as $\left|\delta \right|$ goes to 1; moreover, ${\lambda }_L(0.01;\delta ,\tilde{\rho })$ is higher for $\delta <0$ than for $\delta >0$.

Remark 4 

(Asymptotic formulas of the TDCs). It is shown in [14] that

$$\begin{array}{c}\hfill {\lambda }_L(u;0,\rho )\sim u^{\frac{1-\rho }{1+\rho }}(1+\rho )\sqrt{\frac{1+\rho }{1-\rho }}{(-4\pi logu)}^{-\frac{\rho }{1+\rho }},u\downarrow 0.\end{array}$$

(22)

Moreover, ([15], Theorem 2) shows that, for $u\downarrow 0$,

$$\begin{array}{c}\hfill {\lambda }_L(u;\delta {\mathbf{1}}_2,\rho )\sim \left\{\begin{array}{cc}{u}^{{\beta }^2}\frac{{\left(2\pi \lambda \right)}^{{\beta }^2}}{\sqrt{\pi }\beta {(1+{\beta }^2)}^2}{(-logu)}^{{\beta }^2-\frac{1}{2}},\hfill & \mathrm{if}\delta >0,\hfill \\ u^{\frac{1-\rho }{1+\rho }}\left(\frac{1+\rho }{2}\right)\sqrt{\frac{1+\rho }{1-\rho }}{(-\pi logu)}^{-\frac{\rho }{1+\rho }},\hfill & \mathrm{if}\delta <0,\hfill \end{array}\right.\end{array}$$

(23)

where $\lambda =\zeta =\alpha (1+\rho )/\sqrt{1+{\alpha }^2(1-{\rho }^2)}$, $\alpha ={\alpha }_1={\alpha }_2$, $\zeta ={\zeta }_1={\zeta }_2$ and β is defined in (A2) such that ${\beta }^2+1=2/(1+\tilde{\rho })$.

The following asymptotic formulas of ${\lambda }_L$ are also given in ([16], Appendix B) based on the tail expansion of the skew-normal copula.

$$\begin{array}{c}\hfill {\lambda }_L(u;\delta {\mathbf{1}}_2,\tilde{\rho })\sim \left\{\begin{array}{cc}{\kappa }^{-1}{u}^{\kappa -1}{(-2logu)}^{\kappa -\frac{3}{2}},\hfill & \mathrm{if}\delta >0,\hfill \\ \left(\frac{1+\rho }{2}\right)u^{\frac{1-\rho }{1+\rho }}{(-2logu)}^{-\frac{\rho }{1+\rho }},\hfill & \mathrm{if}\delta <0,\hfill \end{array}\right.\end{array}$$

(24)

where

$$\begin{array}{c}\hfill \kappa =\frac{2(1-{\delta }^2)}{1+\rho -2{\delta }^2}=\frac{2}{1+\tilde{\rho }}.\end{array}$$

Note that the asymptotic formulas in (24) are slightly different from those in (23), but they lead to the same tail order. With this observation, the asymptotic formulas (22),  (23) and (24) yield the same limit (19) by using (20).

4. Accuracy of the asymptotic formulas

In Section 3, various approaches are provided to compute the measure of tail asymmetry of the skew-normal copula. It is partly observed in Figure 3 that the formula (17) with the algorithm TVPACK gives results consistent with the asymptotic formula (19). As mentioned in Remark 4, the measure of tail asymmetry (1) can also be computed from the asymptotic formulas (22),  (23) and (24) derived in the literature. In line with this, this section explores the performance of these asymptotic formulas in a series of numerical experiments. For ease of illustration, we focus on the equi-skewed case ${\delta }_1={\delta }_2=\delta$.

We first compute the value of ${\lambda }_L(u;\delta {\mathbf{1}}_2,\tilde{\rho })$ for $u=0.01$ as an extremely small u by (21) and (11) with ${\mathsf{\Phi }}_3$ evaluated by pmvnorm(algorithm = TVPACK) [12] in the R package mvtnorm. Figure 5 plots the contour of $log{\lambda }_L(0.01;\delta {\mathbf{1}}_2,\tilde{\rho })$ for $\delta \in [-0.999,0.999]$ and $\tilde{\rho }\in [-0.8,0.999]$ by using (11). The range of $\tilde{\rho }$ is restricted due to the numerical limitation. We observe that $log{\lambda }_L(0.01;\delta {\mathbf{1}}_2,\tilde{\rho })$ approaches 0 as $\tilde{\rho }$ and $\left|\delta \right|$ go to 1, which is consistent with the fact that the skew-normal copula is comonotonic for these parameters. Monotonicity of the function ${\lambda }_L(u;\delta {\mathbf{1}}_2,\tilde{\rho })$ with respect to $\delta$ is also indicated from Figure 5. Namely, it is observable that the value of $log{\lambda }_L(0.01;\delta {\mathbf{1}}_2,\tilde{\rho })$ for a fixed $\tilde{\rho }$ is monotonically decreasing for $\delta <0$ and increasing for $\delta >0$, and thus the minimum is attained at $\delta =0$.

For special cases, if $\rho =0$, the correlation between $Y_1$ and $Y_2$ in (2) is given by ${\delta }^2$. If $\delta =\pm 1$, the correlation is one, the variables are comonotonic, and thus $log{\lambda }_L(u;\pm {\mathbf{1}}_2,0)=log1=0$. If $\delta =0$, the variables are independent and thus $log{\lambda }_L(u;{\mathbf{0}}_2,0)=logu$.

Next, we demonstrate the performance of the asymptotic formulas presented in Remark 4. Figure 6 plots the contour of the difference of $log{\lambda }_L(0.01;\delta {\mathbf{1}}_2,\tilde{\rho })$ based on the asymptotic formula (23) and that based on the numerical evaluation of (21) with the algorithm TVPACK. The same plot is provided for the asymptotic formula (24) instead of (23). Interestingly, we observe that the two asymptotic formulas perform well on the different areas of the parameter range. In particular, the difference of the asymptotic approximation based on the formula (23) is large around the boundaries. Discontinuity of the asymptotic formulas is also observed around $\delta =0$, which is more visible for the formula (23).

5. Conclusion

In this paper we explored the measure of tail asymmetry proposed in [9], and its asymptotic behavior. We showed that the measure, after properly scaled, is asymptotically equivalent to the difference of the upper and lower tail orders [10]. Based on this result, we derived an analytical expression of the measure of tail asymmetry for the skew-normal copula. The performance of this formula is verified by comparing it to another analytical formula with a finite threshold. We also investigated the asymptotic formulas of the TDC of the skew-normal copula proposed in the literature. Our numerical experiments revealed that these formulas perform well for moderate values of parameters, but are not recommendable for approximating the measure of tail asymmetry with finite thresholds when the parameters are at their boundaries.

Appendix A.1. Parameters of the bivariate skew-normal copula

In this section we describe detailed derivations of (14) and (16).

For the $2\times 2$ correlation matrices $\mathsf{\Omega }$ and $\mathsf{\Psi }$, let $\rho ={\mathsf{\Omega }}_{12}={\mathsf{\Omega }}_{21}$ and $\tilde{\rho }={\mathsf{\Psi }}_{12}={\mathsf{\Psi }}_{21}$. Then,

$$\rho =\mathrm{cor}\left(Y_1,Y_2\right)=\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\delta }_1{\delta }_2,{\mathsf{\Delta }}_1=\sqrt{1-{\delta }_1^2},{\mathsf{\Delta }}_2=\sqrt{1-{\delta }_2^2}.$$

Here, $\tilde{\rho }$ is the partial correlation of $Y_1$ and $Y_2$ given that $Z_0$ is fixed in the trivariate random vector $\left(Z_0,Y_1,Y_2\right)$ given in (2). By using (5), we have

$$\begin{array}{c}\hfill \mathsf{\Omega }=\mathsf{\Delta }\left(\mathsf{\Psi }+\zeta {\zeta }^{\top }\right)\mathsf{\Delta }=\mathsf{\Delta }\left(\left(\begin{array}{cc}1& \tilde{\rho }\\ \tilde{\rho }& 1\end{array}\right)+\left(\begin{array}{cc}{\zeta }_1^2& {\zeta }_1{\zeta }_2\\ {\zeta }_1{\zeta }_2& {\zeta }_2^2\end{array}\right)\right)\mathsf{\Delta },\end{array}$$

where

$$\begin{array}{c}\hfill \mathsf{\Delta }=\left(\begin{array}{cc}{\mathsf{\Delta }}_1& 0\\ 0& {\mathsf{\Delta }}_2\end{array}\right)=\left(\begin{array}{cc}\sqrt{1-{\delta }_1^2}& 0\\ 0& \sqrt{1-{\delta }_2^2}\end{array}\right)=\left(\begin{array}{cc}1/\sqrt{1+{\zeta }_1^2}& 0\\ 0& 1/\sqrt{1+{\zeta }_2^2}\end{array}\right).\end{array}$$

Hence, we have that

$$\begin{array}{cc}\hfill \mathsf{\Omega }& =\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)=\mathsf{\Delta }\left(\begin{array}{cc}1+{\zeta }_1^2& \tilde{\rho }+{\zeta }_1{\zeta }_2\\ \tilde{\rho }+{\zeta }_1{\zeta }_2& 1+{\zeta }_2^2\end{array}\right)\mathsf{\Delta }\hfill \\ \hfill & =\left(\begin{array}{cc}1& \frac{\tilde{\rho }+{\zeta }_1{\zeta }_2}{\sqrt{1+{\zeta }_1^2}\sqrt{1+{\zeta }_2^2}}\\ \frac{\tilde{\rho }+{\zeta }_1{\zeta }_2}{\sqrt{1+{\zeta }_1^2}\sqrt{1+{\zeta }_2^2}}& 1\end{array}\right)=\left(\begin{array}{cc}1& \tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\delta }_1{\delta }_2\\ \tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\delta }_1{\delta }_2& 1\end{array}\right),\hfill \end{array}$$

which yields (14).

Next we check $\alpha$. By using

$$\begin{array}{cc}\hfill & {\mathsf{\Omega }}^{-1}\delta =\frac{1}{1-{\rho }^2}\left(\begin{array}{cc}1& -\rho \\ -\rho & 1\end{array}\right)\left(\begin{array}{c}{\delta }_1\\ {\delta }_2\end{array}\right)=\frac{1}{1-{\rho }^2}\left(\begin{array}{c}{\delta }_1-\rho {\delta }_2\\ {\delta }_2-\rho {\delta }_1\end{array}\right),\hfill \\ \hfill & 1-{\delta }^{\top }{\mathsf{\Omega }}^{-1}\delta =\frac{1-{\rho }^2-{\delta }_1^2+2\rho {\delta }_1{\delta }_2-{\delta }_2^2}{1-{\rho }^2},\hfill \end{array}$$

we have from () that

$$\alpha =\frac{{\mathsf{\Omega }}^{-1}\delta }{\sqrt{1-{\delta }^{\top }{\mathsf{\Omega }}^{-1}\delta }}=\frac{1}{\sqrt{\left(1-{\rho }^2\right)\left(1-{\rho }^2-{\delta }_1^2+2\rho {\delta }_1{\delta }_2-{\delta }_2^2\right)}}\left(\begin{array}{c}{\delta }_1-\rho {\delta }_2\\ {\delta }_2-\rho {\delta }_1\end{array}\right).$$

(A1)

Recall that the extended correlation matrix ${\mathsf{\Omega }}^{*}\left(\delta \right)$ in (9) is given by

$${\mathsf{\Omega }}^{*}\left(\delta \right)=\left(\begin{array}{ccc}1& {\delta }_1& {\delta }_2\\ {\delta }_1& 1& \rho \\ {\delta }_2& \rho & 1\end{array}\right).$$

Since

$$det{\mathsf{\Omega }}^{*}\left(\delta \right)=1-{\rho }^2-{\delta }_1^2+2\rho {\delta }_1{\delta }_2-{\delta }_2^2,$$

the parameter $\alpha$ in (A1) is given by

$$\alpha =\left(\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right)=\frac{1}{\sqrt{\left(1-{\rho }^2\right)det{\mathsf{\Omega }}^{*}\left(\delta \right)}}\left(\begin{array}{c}{\delta }_1-\rho {\delta }_2\\ {\delta }_2-\rho {\delta }_1\end{array}\right).$$

Note that this representation coincides with that in [16, Appendix B], where $({\alpha }_1,{\alpha }_2)$ and ${\mathsf{\Omega }}^{*}\left(\delta \right)$ are denoted by $({\beta }_1,{\beta }_2)$ and $\mathbf{R}$, respectively. Since

$$\begin{array}{cc}\hfill det{\mathsf{\Omega }}^{*}\left(\delta \right)=& 1-{\rho }^2-{\delta }_1^2+2\rho {\delta }_1{\delta }_2-{\delta }_2^2\hfill \\ \hfill & =1-{\left(\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\delta }_1{\delta }_2\right)}^2-{\delta }_1^2+2\left(\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\delta }_1{\delta }_2\right){\delta }_1{\delta }_2-{\delta }_2^2\hfill \\ \hfill & =1+{\delta }_1^2{\delta }_2^2-{\tilde{\rho }}^2\left(1-{\delta }_1^2\right)\left(1-{\delta }_2^2\right)-{\delta }_1^2-{\delta }_2^2\hfill \\ \hfill & =\left(1-{\tilde{\rho }}^2\right)\left(1-{\delta }_1^2\right)\left(1-{\delta }_2^2\right),\hfill \end{array}$$

we obtain (16).

Appendix A.2. Tail orders of the bivariate skew-normal copula

In this section we summarize tail orders of the bivariate skew-normal copula known in the literature. Since ${\kappa }_U(C_{SN};\delta ,\tilde{\rho })={\kappa }_L(C_{SN};-\delta ,\tilde{\rho })$, we study only the lower tail order ${\kappa }_L(C_{SN};\delta ,\tilde{\rho })$, $\tilde{\rho }\in (-1,1)$, for various $\delta =({\delta }_1,{\delta }_2)\in {(-1,1)}^2$. Moreover, we focus on the case when ${\delta }_1$ and ${\delta }_2$ have the same sign, that is, either ${\delta }_1,{\delta }_2\ge 0$ or ${\delta }_1,{\delta }_2\le 0$. In this case, we will show below that the lower tail order is summarized by (18). The interested reader is referred to [13] for more general cases of the skewness parameter. Note that the explicit forms of the tail orders of the skew-normal copula can also be found, for example, in [13,15,16]. Although they cover different cases of the parameters, their results are consistent with each other.

Case I: δ 1 =δ 2 =δ

We first consider the equi-skewed case ${\delta }_1={\delta }_2=\delta \in (-1,1)$. It follows from ([15], Theorem 2) that

$$\begin{array}{c}\hfill {\kappa }_L(C_{SN};\delta ,\tilde{\rho })=\left\{\begin{array}{cc}{\beta }^2+1,\hfill & \mathrm{if}\alpha >0,\hfill \\ \frac{2}{1+\rho },\hfill & \mathrm{if}\alpha <0,\hfill \end{array}\right.\end{array}$$

where

$$\begin{array}{c}\hfill \alpha =\frac{\delta }{\sqrt{(1+\tilde{\rho })(1+\rho )(1-{\delta }^2)}},\end{array}$$

and

$$\begin{array}{c}\hfill \beta =\sqrt{\frac{(1-\rho )(1+2(1+\rho ){\alpha }^2)}{1+\rho }},\end{array}$$

(A2)

and thus, by calculation,

$$\begin{array}{c}\hfill {\beta }^2+1=\frac{1-\tilde{\rho }}{1+\tilde{\rho }}+1=\frac{2}{1+\tilde{\rho }}.\end{array}$$

Note that the signs of $\alpha$ and $\delta$ are identical, and that $\delta =0$ if and only if $\alpha =0$. When $\delta =0$, we have that

$$\begin{array}{c}\hfill {\kappa }_L(C_{SN};\mathbf{0},\tilde{\rho })=\frac{2}{1+\tilde{\rho }}=\frac{2}{1+\rho };\end{array}$$

see also (Li and Joe [16], Appendix B).

Case II: δ 1 ,δ 2 <0

It is shown in [13] that

$${\kappa }_L\left(C_{\mathrm{SN}};\delta ,\tilde{\rho }\right)=\frac{2}{1+\rho },$$

which is also derived in [16]. Note that the condition ${\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2<0$ is imposed in [16], and this condition is implied by (15). Indeed, we have from (16) that

$${\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2=\frac{{\mathsf{\Delta }}_1\left({\delta }_1-\rho {\delta }_2\right)+{\mathsf{\Delta }}_2\left({\delta }_2-\rho {\delta }_1\right)}{\sqrt{\left(1-{\rho }^2\right)det{\mathsf{\Omega }}^{*}\left(\delta \right)}},$$

(A3)

and thus the sign of ${\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2$ equals to that of the numerator of the right-hand side of (A3). From (15), the inequality ${\mathsf{\Delta }}_1{\delta }_2+{\mathsf{\Delta }}_2{\delta }_1<0$ implies that

$$\begin{array}{cc}\hfill & {\mathsf{\Delta }}_1\left({\delta }_1-\rho {\delta }_2\right)+{\mathsf{\Delta }}_2\left({\delta }_2-\rho {\delta }_1\right)={\mathsf{\Delta }}_1{\delta }_1+{\mathsf{\Delta }}_2{\delta }_2-\rho \left({\mathsf{\Delta }}_1{\delta }_2+{\mathsf{\Delta }}_2{\delta }_1\right)\hfill \\ \hfill & <{\mathsf{\Delta }}_1{\delta }_1+{\mathsf{\Delta }}_2{\delta }_2-\left({\delta }_1{\delta }_2+{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\right)\left({\mathsf{\Delta }}_1{\delta }_2+{\mathsf{\Delta }}_2{\delta }_1\right)\hfill \\ \hfill & ={\mathsf{\Delta }}_1{\delta }_1+{\mathsf{\Delta }}_2{\delta }_2-{\mathsf{\Delta }}_1{\delta }_1{\delta }_2^2-{\mathsf{\Delta }}_2{\delta }_1^2{\delta }_2-{\mathsf{\Delta }}_2\left({1-\delta }_1^2\right){\delta }_2-{\mathsf{\Delta }}_1\left({1-\delta }_2^2\right){\delta }_1=0,\hfill \end{array}$$

and thus ${\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2<0$.

Case III: δ 1 ,δ 2 >0

It is shown in [13] that

$$\begin{array}{cc}\hfill & {\kappa }_L\left(C_{\mathrm{SN}}:\delta ,\tilde{\rho }\right)={\kappa }_U\left(C_{\mathrm{SN}}:-\delta ,\tilde{\rho }\right)\hfill \\ \hfill & =\frac{1}{1-{\rho }^2}\left\{\frac{1+{\alpha }_1^2\left(1-{\rho }^2\right)}{1+{\zeta }_1^2}+\frac{1+{\alpha }_2^2\left(1-{\rho }^2\right)}{1+{\zeta }_2^2}+\frac{2\left({\alpha }_1{\alpha }_2\left(1-{\rho }^2\right)-\rho \right)}{\sqrt{\left(1+{\zeta }_1^2\right)\left(1+{\zeta }_2^2\right)}}\right\}\hfill \\ \hfill & =\frac{1}{1-{\rho }^2}\left\{\frac{1}{1+{\zeta }_1^2}+\frac{1}{1+{\zeta }_2^2}-\frac{2\rho }{\sqrt{\left(1+{\zeta }_1^2\right)\left(1+{\zeta }_2^2\right)}}\right.\hfill \\ \hfill & \left.+\left(1-{\rho }^2\right){\left(\frac{{\alpha }_1}{\sqrt{1+{\zeta }_1^2}}+\frac{{\alpha }_2}{\sqrt{1+{\zeta }_2^2}}\right)}^2\right\}\hfill \\ \hfill & =\frac{{\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2-2\rho {\mathsf{\Delta }}_1{\mathsf{\Delta }}_2}{1-{\rho }^2}+{\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2.\hfill \end{array}$$

(A4)

Note that, in [13], ${\zeta }_1$ and ${\zeta }_2$ above are denoted by ${\lambda }_1$ and ${\lambda }_2$, respectively. The formula (A4) is also derived in Appendix B of [16]. Note again that the condition ${\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2>0$ imposed in [16] is implied by (15). We will check that

$$\begin{array}{c}\hfill \frac{{\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2-2\rho {\mathsf{\Delta }}_1{\mathsf{\Delta }}_2}{1-{\rho }^2}+{\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2=\frac{2}{1+\tilde{\rho }},\end{array}$$

(A5)

that is, the expression (A4) can be simplified as

$${\kappa }_L\left(C_{\mathrm{SN}}:\delta ,\tilde{\rho }\right)={\kappa }_U\left(C_{\mathrm{SN}}:-\delta ,\tilde{\rho }\right)=\frac{2}{1+\tilde{\rho }}.$$

First, by multiplying $\left(1-{\rho }^2\right)$ on the right-hand side of (A5), the desired equation is equivalent to

$${\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2-2\rho {\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+\left(1-{\rho }^2\right){\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2=\frac{2\left(1-\tilde{\rho }\right)\left(1-{\rho }^2\right)}{\left(1-{\tilde{\rho }}^2\right)},$$

which is also equivalent to

$$\begin{array}{cc}\hfill & (1-{\tilde{\rho }}^2){\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2\left\{{\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2-2\rho {\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+\left(1-{\rho }^2\right){\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2\right\}=2\left(1-\tilde{\rho }\right)\left(1-{\rho }^2\right){\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2\hfill \end{array}$$

(A6)

by multiplying $\left(1-{\tilde{\rho }}^2\right){\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2$ on both sides. Since

$${\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2=\frac{{\left\{\left({\delta }_1-\rho {\delta }_2\right){\mathsf{\Delta }}_1+\left({\delta }_2-\rho {\delta }_1\right){\mathsf{\Delta }}_2\right\}}^2}{\left(1-{\rho }^2\right)\left(1-{\tilde{\rho }}^2\right)\left(1-{\delta }_1^2\right)\left(1-{\delta }_2^2\right)},$$

the left-hand side of (A6) reduces to

$$\begin{array}{c}\hfill (1-{\tilde{\rho }}^2){\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2({\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2)-2\rho (1-{\tilde{\rho }}^2){\mathsf{\Delta }}_1^3{\mathsf{\Delta }}_2^3+{\left\{\left({\delta }_1-\rho {\delta }_2\right){\mathsf{\Delta }}_1+\left({\delta }_2-\rho {\delta }_1\right){\mathsf{\Delta }}_2\right\}}^2.\end{array}$$

(A7)

The last two terms of (A7) are expanded into

$$\begin{array}{cc}\hfill -2\rho \left(1-{\tilde{\rho }}^2\right){\mathsf{\Delta }}_1^3{\mathsf{\Delta }}_2^3& +{\left\{\left({\delta }_1-\rho {\delta }_2\right){\mathsf{\Delta }}_1+\left({\delta }_2-\rho {\delta }_1\right){\mathsf{\Delta }}_2\right\}}^2\hfill \\ \hfill & =-2\rho \left(1-{\tilde{\rho }}^2\right){\mathsf{\Delta }}_1^3{\mathsf{\Delta }}_2^3+2\left({\delta }_1-\rho {\delta }_2\right)\left({\delta }_2-\rho {\delta }_1\right){\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\hfill \\ \hfill & +{\left({\delta }_1-\rho {\delta }_2\right)}^2{\mathsf{\Delta }}_1^2+{\left({\delta }_2-\rho {\delta }_1\right)}^2{\mathsf{\Delta }}_2^2.\hfill \end{array}$$

(A8)

By using ${\delta }_1{\delta }_2-\rho =-\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2$, the coefficient of ${\mathsf{\Delta }}_1{\mathsf{\Delta }}_2$ at the first two terms in the right-hand side of (A8) is given by

$$\begin{array}{cc}\hfill & -2\rho \left(1-{\tilde{\rho }}^2\right)\left(1-{\delta }_1^2\right)\left(1-{\delta }_2^2\right)+2\left({\delta }_1-\rho {\delta }_2\right)\left({\delta }_2-\rho {\delta }_1\right)\hfill \\ \hfill & =-2\rho \left(1-{\delta }_1^2-{\delta }_2^2+{\delta }_1^2{\delta }_2^2\right)+2\rho {\tilde{\rho }}^2\left(1-{\delta }_1^2\right)\left(1-{\delta }_2^2\right)+2\left({\delta }_1{\delta }_2-\rho {\delta }_1^2-\rho {\delta }_2^2+{\rho }^2{\delta }_1{\delta }_2\right)\hfill \\ \hfill & =2\left({\delta }_1{\delta }_2-\rho -\rho {\delta }_1^2{\delta }_2^2+{\rho }^2{\delta }_1{\delta }_2\right)+2\rho {\tilde{\rho }}^2\left(1-{\delta }_1^2\right)\left(1-{\delta }_2^2\right)\hfill \\ \hfill & =-2\tilde{\rho }\left(1-\rho {\delta }_1{\delta }_2\right){\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+2\rho {\tilde{\rho }}^2{\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2\hfill \\ \hfill & =-2\tilde{\rho }\left(1-\rho {\delta }_1{\delta }_2-\rho \tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\right){\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\hfill \\ \hfill & =-2\tilde{\rho }\left(1-{\rho }^2\right){\mathsf{\Delta }}_1{\mathsf{\Delta }}_2.\hfill \end{array}$$

(A9)

By using $\rho =\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+{\delta }_1{\delta }_2$, the last two terms of the right-hand side of (A8) are rearranged as follows:

$$\begin{array}{cc}\hfill & {\left({\delta }_1-\rho {\delta }_2\right)}^2{\mathsf{\Delta }}_1^2+{\left({\delta }_2-\rho {\delta }_1\right)}^2{\mathsf{\Delta }}_2^2\hfill \\ \hfill & ={\left({\delta }_1\left(1-{\delta }_2^2\right)-\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2{\delta }_2\right)}^2{\mathsf{\Delta }}_1^2+{\left({\delta }_2\left(1-{\delta }_1^2\right)-\tilde{\rho }{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2{\delta }_1\right)}^2{\mathsf{\Delta }}_2^2\hfill \\ \hfill & =\left\{{({\delta }_1{\mathsf{\Delta }}_2-\tilde{\rho }{\mathsf{\Delta }}_1{\delta }_2)}^2+{({\delta }_2{\mathsf{\Delta }}_1-\tilde{\rho }{\mathsf{\Delta }}_2{\delta }_1)}^2\right\}{\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2\hfill \\ \hfill & =\left\{\left({\delta }_1^2+{\delta }_2^2-2{\delta }_1^2{\delta }_2^2\right)\left(1+{\tilde{\rho }}^2\right)-4\tilde{\rho }{\delta }_1{\delta }_2{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\right\}{\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2.\hfill \end{array}$$

(A10)

Combining (A9) and (A10), the term (A8) reduces to

$$\begin{array}{cc}\hfill -2\rho & \left(1-{\tilde{\rho }}^2\right){\mathsf{\Delta }}_1^3{\mathsf{\Delta }}_2^3+{\left\{\left({\delta }_1-\rho {\delta }_2\right){\mathsf{\Delta }}_1+\left({\delta }_2-\rho {\delta }_1\right){\mathsf{\Delta }}_2\right\}}^2\hfill \\ \hfill & =\left\{-2\tilde{\rho }\left(1-{\rho }^2\right)+\left({\delta }_1^2+{\delta }_2^2-2{\delta }_1^2{\delta }_2^2\right)\left(1+{\tilde{\rho }}^2\right)-4\tilde{\rho }{\delta }_1{\delta }_2{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\right\}{\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2.\hfill \end{array}$$

Therefore, the desired equation (A6) is now

$$\begin{array}{c}\hfill (1-{\tilde{\rho }}^2)\left\{{\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2-2\rho {\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+\left(1-{\rho }^2\right){\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2\right\}=2\left(1-\tilde{\rho }\right)\left(1-{\rho }^2\right),\end{array}$$

which can be checked as follows:

$$\begin{array}{cc}\hfill & (1-{\tilde{\rho }}^2)\left\{{\mathsf{\Delta }}_1^2+{\mathsf{\Delta }}_2^2-2\rho {\mathsf{\Delta }}_1{\mathsf{\Delta }}_2+\left(1-{\rho }^2\right){\left({\mathsf{\Delta }}_1{\alpha }_1+{\mathsf{\Delta }}_2{\alpha }_2\right)}^2\right\}\hfill \\ \hfill & =\left(2-{\delta }_1^2-{\delta }_2^2\right)\left(1-{\tilde{\rho }}^2\right)-2\tilde{\rho }\left(1-{\rho }^2\right)+\left({\delta }_1^2+{\delta }_2^2-2{\delta }_1^2{\delta }_2^2\right)\left(1+{\tilde{\rho }}^2\right)-4\tilde{\rho }{\delta }_1{\delta }_2{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\hfill \\ \hfill & =2\left(1-{\delta }_1^2{\delta }_2^2-{\tilde{\rho }}^2{\mathsf{\Delta }}_1^2{\mathsf{\Delta }}_2^2-2\tilde{\rho }{\delta }_1{\delta }_2{\mathsf{\Delta }}_1{\mathsf{\Delta }}_2\right)-2\tilde{\rho }\left(1-{\rho }^2\right)\hfill \\ \hfill & =2(1-{\rho }^2)-2\tilde{\rho }(1-{\rho }^2)\hfill \\ \hfill & =2\left(1-\tilde{\rho }\right)\left(1-{\rho }^2\right).\hfill \end{array}$$

Case IV: One of δ 1 and δ 2 is zero and the other is negative

By symmetry, it suffices to consider the case when ${\delta }_1=0$ and ${\delta }_2<0$. In this case, [13] shows that ${\kappa }_L\left(C_{\mathrm{SN}}:\delta ,\tilde{\rho }\right)=2/(1+\rho )$.

Case V: One of δ 1 and δ 2 is zero and the other is positive

By symmetry, it suffices to consider the case when ${\delta }_1=0$ and ${\delta }_2>0$. In this case, [13] shows that, if ${\alpha }_1+{\alpha }_2{\mathsf{\Delta }}_2>0$, then

$$\begin{array}{c}\hfill {\kappa }_L\left(C_{\mathrm{SN}}:\delta ,\tilde{\rho }\right)=\frac{{({\mathsf{\Delta }}_2-\rho )}^2}{(1-{\rho }^2)}+{({\alpha }_1+{\alpha }_2{\mathsf{\Delta }}_2)}^2+1.\end{array}$$

The condition ${\alpha }_1+{\alpha }_2{\mathsf{\Delta }}_2>0$ is always satisfied since

$$\begin{array}{c}\hfill {\alpha }_1+{\alpha }_2{\mathsf{\Delta }}_2=\frac{{\delta }_2({\mathsf{\Delta }}_2-\rho )}{\sqrt{(1-{\rho }^2)det{\mathsf{\Omega }}^{*}\left(\delta \right)}}\end{array}$$

and $\rho <{\mathsf{\Delta }}_2$ by (15). Since $det{\mathsf{\Omega }}^{*}\left(\delta \right)=(1-{\tilde{\rho }}^2)(1-{\delta }_2^2)$ and $\rho =\tilde{\rho }{\mathsf{\Delta }}_2$, we have that

$$\begin{array}{cc}\hfill \frac{{({\mathsf{\Delta }}_2-\rho )}^2}{(1-{\rho }^2)}+{({\alpha }_1+{\alpha }_2{\mathsf{\Delta }}_2)}^2& =\frac{{({\mathsf{\Delta }}_2-\rho )}^2\{{\delta }_2^2+det{\mathsf{\Omega }}^{*}\left(\delta \right)\}}{(1-{\rho }^2)det{\mathsf{\Omega }}^{*}\left(\delta \right)}\hfill \\ \hfill & =\frac{{(1-\tilde{\rho })}^2(1-{\tilde{\rho }}^2+{\tilde{\rho }}^2{\delta }_2^2)}{(1-{\rho }^2)(1-{\tilde{\rho }}^2)}\hfill \\ \hfill & =\frac{(1-\tilde{\rho })(1-{\tilde{\rho }}^2(1-{\delta }_2^2))}{(1-{\rho }^2)(1+\tilde{\rho })}\hfill \\ \hfill & =\frac{1-\tilde{\rho }}{1+\tilde{\rho }}.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\kappa }_L\left(C_{\mathrm{SN}}:\delta ,\tilde{\rho }\right)=\frac{{({\mathsf{\Delta }}_2-\rho )}^2}{(1-{\rho }^2)}+{({\alpha }_1+{\alpha }_2{\mathsf{\Delta }}_2)}^2+1=\frac{2}{1+\tilde{\rho }}.\end{array}$$