Tail asymptotics for the bivariate equi-skew Variance-Gamma distribution

The coefficient of lower tail dependence of a random vector $\textbf{X}=(X_{1},X_{2})^{\top}$ with marginal inverse distribution functions $F_{1}^{-1}$ and $F_{2}^{-1}$ is defined as

This quantity provides insight on the tendency for the distribution to generate joint extreme event since it measures the strength of dependence (or association) in the lower tail of a bivariate distribution. If the marginal distributions of these random variables are continuous, then from (1), it follows that $\lambda_{L}(u)$ can be expressed in terms of the copula of X, $C(u_{1},u_{2})$, as

If $\lambda_{L}=0$ in (1), that is, if asymptotic lower tail independence obtains, the asymptotic rate of convergence to zero as $u\to 0^{+}$ of the copula $C(u,u)$ is tantamount to that of $\lambda_{L}(u)$ through the relation:

The central purpose of this paper is to provide an analytic result on the asymptotic tail independence for the bivariate skew Variance-Gamma (VG) model. We will consider this problem in terms of the more general skew Generalized Hyperbolic (GH) distribution.

The (standardized) bivariate skew GH distribution, $GH(0,R,\boldsymbol{\theta},p,a,b)$ is defined by its variance-mean mixing representation as

where $\textbf{X}=(X_{1},X_{2})^{\top},\boldsymbol{\theta}^{\top}=(\theta_{1},\theta_{2})$, and $W\sim GIG(p,a,b)$ is independently distributed of $\textbf{Z}\sim N(0,R)$. Here $R=\left(\begin{smallmatrix}1&&\rho\\ \\ \rho&&1\end{smallmatrix}\right)$, with $-1<\rho<1$.

Recall that a random variable $W$ is said to have a (univariate) Generalised Inverse Gaussian (GIG) distribution, denoted by $GIG(p,a,b)$, if it has density

where

Here $K_{p}(\omega),\ \omega>0,$ is the modified Bessel function of the second kind (Erdélyi \BOthers. (\APACyear1954)) with index $p\in\mathbb{R}$.

In the VG special case $a=0$, $b=\sqrt{\frac{2}{\nu}}$, $p=\frac{1}{2}$. We proceed more generally by assuming $b>0$ in this note in the $GIG(p,a,b)$ setting.

It was shown in Fung \BBA Seneta (\APACyear2011) that when $\textbf{X}\sim GH(0,R,\boldsymbol{\theta},p,a,b)$ with $b>0$, then X is asymptotically independent in the lower tail; that is $\lambda_{L}=0.$ The proof in von Hammerstein (\APACyear2016) for the VG can be adjusted to give this same conclusion.

Our specific focus in the sequel is to obtain a rate of convergence result of the form:

where $L(u)$ is a slowly varying function (SVF) as $u\to 0^{+}$ and $\tau>1,$ when $\theta_{1}=\theta_{2},=\theta$, say, so $X_{i}\sim\theta W+\sqrt{W}Z_{i}$, $i=1,2$, where $Z_{i}\sim N(0,1).$ That is, the distribution functions of the $X_{i},i=1,2$ are the same: $F_{1}(u)=F_{2}(u),=F(u)$, say. We call this assumption in a bivariate setting “equi-skewness”.

Our study thus parallels that of Fung \BBA Seneta (\APACyear2016), who treat the bivariate skew normal distributed X, that is $\textbf{X}\sim SN_{2}({\bf\alpha},R)$ where in ${\bf\alpha}=(\alpha_{1},\alpha_{2})^{\top}$, it is assumed that $\alpha_{1}=\alpha_{2},=\alpha$ say, so equi-skewness obtains.

Both treatments depend on the same initial device: that

to reduce the bivariate problem to a univariate one. Our subsequent treatment is quite different, since the setting in Fung \BBA Seneta (\APACyear2016) is just mean-mixing, and with a mixing distribution not encompassed by the GIG.

Clearly, since $\lambda_{L}(u)$ is a probability, the index $\tau$ in (4) must satisfy $\tau\geq 1.$ We note that Ledford \BBA Tawn (\APACyear1997), Ramos \BBA Ledford (\APACyear2009), Hashorva (\APACyear2010) and Hua \BBA Joe (\APACyear2011) all classified the degree of tail-dependence to the value of $\tau$ in (4). Hua \BBA Joe (\APACyear2011) define $\tau$ in (4) as the (lower) tail order of a copula. The tail order case $1<\tau<2$ is considered as intermediate tail dependence as it corresponds to the copula having some level of positive dependence in the tail when $\lambda_{L}=0$. Thus when $\lambda_{L}(u)=C(u,u)/u=u^{\tau-1}L(u)$, $1<\tau<2$, there is some measure of positive association when $\lambda_{L}=0$, but the association is not as strong as when $\tau=1$, and $\lambda_{L}(u)=L(u)\to\lambda_{L}>0$, $u\to 0^{+}$, the case of asymptotic tail dependence. In our specific setting we shall find that $1<\tau<\infty.$

For the case of $b=0$ in the $GIG(p,a,b)$ setting, X can be asymptotically dependent in the lower tail. The limiting and rate of convergence results for this case were discussed in Fung \BBA Seneta (\APACyear2010) and Fung \BBA Seneta (\APACyear2014) respectively.