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]: 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 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 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.

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.

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.

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).

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.