3.2 What is Tail Dependence?

Definitions of tail dependence for multivariate random vectors are mostly related to their bivariate marginal distribution functions. Loosely speaking, tail dependence describes the limiting proportion that one margin exceeds a certain threshold given that the other margin has already exceeded that threshold. The following approach, as provided in the monograph of Joe (1997), represents one of many possible definitions of tail dependence.

Let $ X=(X_1,X_2)^\top$ be a two-dimensional random vector. We say that $ X$ is (bivariate) upper tail-dependent if:

$\displaystyle \lambda_U\stackrel{\mathrm{def}}{=}\lim_{v\uparrow 1} <tex2html_comment_mark>438 \textrm{P}\left\{X_1>F^{-1}_1(v)\mid X_2>F^{-1}_2(v)\right\}>0,$ (3.1)

in case the limit exists. $ F^{-1}_1$ and $ F^{-1}_2$ denote the generalized inverse distribution functions of $ X_1$ and $ X_2,$ respectively. Consequently, we say $ X=(X_1,X_2)^\top$ is upper tail-independent if $ \lambda_U$ equals $ 0.$ Further, we call $ \lambda_U$ the upper tail-dependence coefficient (upper TDC). Similarly, we define the lower tail-dependence coefficient, if it exists, by:

% latex2html id marker 48888
$\displaystyle \lambda_L\stackrel{\mathrm{def}}{=}\...
...wnarrow 0}\textrm{P}\left\{X_1\leq F^{-1}_1(v)\mid X_2\leq F^{-1}_2(v)\right\}.$ (3.2)

In case $ X=(X_1,X_2)^\top$ is standard normally or $ t$-distributed, formula (3.1) simplifies to:

$\displaystyle \lambda_U=\lim_{v\uparrow 1}\lambda_U(v)\stackrel{\mathrm{def}}{=...
...\uparrow 1} 2\cdot\textrm{P}\left\{X_1>F^{-1}_1(v)\mid X_2=F^{-1}_2(v)\right\}.$ (3.3)

A generalization of bivariate tail dependence, as defined above, to multivariate tail dependence can be found in Schmidt and Stadtmüller (2003).

Figure 3.1: The function $ \lambda _U(v)=2\cdot \textrm {P}\{X_1>F^{-1}_1(v)\mid X_2=F^{-1}_2(v)\}$ for a bivariate normal distribution with correlation coefficients $ \rho=-0.8,\;
-0.6,\dots,0.6,\;0.8.$ Note that $ \lambda _U=0$ for all $ \rho \in (-1,1).$

Figure 3.2: The function $ \lambda _U(v)=2\cdot \textrm {P}\{X_1>F^{-1}_1(v)\mid X_2=F^{-1}_2(v)\}$ for a bivariate $ t$-distribution with correlation coefficients $ \rho=-0.8,\; -0.6,\dots,0.6,$ $ 0.8.$

Figures 3.1 and 3.2 illustrate tail dependence for a bivariate normal and $ t$-distribution. Irrespectively of the correlation coefficient $ \rho,$ the bivariate normal distribution is (upper) tail independent. In contrast, the bivariate $ t$-distribution exhibits (upper) tail dependence and the degree of tail dependence is affected by the correlation coefficient $ \rho.$

The concept of tail dependence can be embedded within the copula theory. An $ n$-dimensional distribution function $ C:[0,1]^n\rightarrow
[0,1]$ is called a copula if it has one-dimensional margins which are uniformly distributed on the interval $ [0,1].$ Copulae are functions that join or ``couple'' an $ n$-dimensional distribution function $ F$ to its corresponding one-dimensional marginal distribution functions $ F_i,\; i=1,\dots,n$, in the following way:

$\displaystyle F(x_1,\dots,x_n)=C\left\{F_1(x_1),\dots,F_n(x_n)\right\}.
$

We refer the reader to Joe (1997), Nelsen (1999) or Härdle, Kleinow, and Stahl (2002) for more information on copulae. The following representation shows that tail dependence is a copula property. Thus, many copula features transfer to the tail-dependence coefficient such as the invariance under strictly increasing transformations of the margins. If $ X$ is a continuous bivariate random vector, then straightforward calculation yields:

$\displaystyle \lambda_U=\lim_{v\uparrow 1}\frac{1-2v+C(v,v)}{1-v},$ (3.4)

where $ C$ denotes the copula of $ X.$ Analogously, % latex2html id marker 48966
$ \lambda_L=\lim_{v\downarrow 0}\frac{C(v,v)}{v}$ holds for the lower tail-dependence coefficient.