Tail dependence coefficients¶

The tail dependence coefficients helps to assess the asymptotic dependency of a bivariate random variables. We detail here the following ones:

the upper tail dependence coefficient denoted by $\lambda_U$ (which is the notation of the probability community) as well as $\chi$ (which is the notation of the extreme value community),
the upper extremal dependence coefficient denoted by $\bar{\chi}$ ,
the lower tail dependence coefficient denoted by $\lambda_L$ ,
the upper extremal dependence coefficient denoted by $\bar{\chi}_L$ .

Readers should refer to [beirlant2004] to get more details.

Let $\vect{X} = (X_1, X_2)$ be a bivariate random vector with marginal distribution functions $F_1$ and $F_2$ , and copula $C$ .

Upper tail dependence coefficient

We denote by $\lambda_U$ or $\chi$ the upper tail dependence coefficient:

$\lambda_U = \chi = \lim_{u \to 1} \Pset[F_2(X_2) > u | F_1(X_1) > u]$

provided that the limit exists.

The $\chi$ coefficient can be interpreted as the tendency for one variable to take extreme high values given that the other variable is extremely high.

The variables $(X_1, X_2)$ are said to be:

asymptotically independent if and only if $\chi=0$ ,
asymptotically dependent if and only if $0 <\chi \leq 1$ .

Now, we define the function $\chi(u)$ defined by:

$\chi(u) = 2 - \frac{\log C(u,u)}{\log u}, \forall u \in [0,1]$

We can see the tail dependence coefficient as the limit of the function $\chi(u)$ when $u$ tends to $1$ . As a matter of fact, when $u$ is close to $1$ , we have:

$\chi(u) = 2 - \frac{1-C(u,u)}{1-u} + o(1) = \Pset[F_2(X_2) > u | F1(X_1) > u] + o(1)$

which proves that:

$\lim_{u \to 1} \chi(u) = \lambda_U = \chi$

Next to providing the limit $\chi$ , the function $\chi(u)$ also provides some insight in the dependence structure of the variables at lower quantile levels: the larger $|\chi(u)|$ the more correlated are the variables.

The function $\chi(u)$ is estimated on data for several $u$ which must not be too high because of a problem of lack of data greater than $u$ . Condifence intervals can be estimated on each $\chi(u)$ . If the confidence interval contains the zero value when $u$ tends to $1$ , then we can assume that $\chi=0$ .

Within the class of asymptotically dependent variables, the value of $\chi$ increases with increasing degree of dependence at extreme levels.

We illustrate two cases where the variables are:

asymptotically independent: we generated a sample of size $10^3$ from a Frank copula which has a zero upper tail coefficient whatever the parameter,
asymptotically dependent: we generated a sample of size $10^3$ from a Gumbel copula parametrized by $\theta = 2.0$ which has a positive upper tail coefficient.

(Source code, png)

Upper extremal dependence coefficient

Within the class of asymptotically independent variables, the degrees of relative strength of dependence is given by the function $\chi(u)$ defined by:

$\bar{\chi}(u) = \frac{\log (\Pset [F_1(X_1) > u] \Pset [F_2(X_2) > u])}{\log [F_1(X_1) > u, F_2(X_2) > u]}, \forall u \in [0,1]$

We show that:

$\bar{\chi}(u) = \frac{2 \log 1-u}{\log \bar{C}(u,u)} - 1, \forall u \in [0,1]$

where $\bar{C}$ is the copula survivor function defined by:

$\bar{C}(u_1, u_2) = \Pset [U_1 > u_1, U_2 > u_2] = 1-u_1-u_2+C(u_1, u_2), \forall u \in [0,1]$

And we can define the upper extremal dependence coefficient by:

$\bar{\chi} = \lim_{u \to 1} \bar{\chi}(u)$

We show that $-1 \leq \bar{\chi} \leq 1$ and that if the variables are asymptotically dependent, then $\bar{\chi} =1$ :

$\chi > 0 \Rightarrow \lim_{u \to 1} \bar{\chi}(u) = 1$

We illustrate the function $\bar{\chi}(u)$ for both previous cases.

(Source code, png)

As a result, the pair $(\chi, \bar{\chi})$ can be used as a summary of extremal dependence of $\vect{X} = (X_1, X_2)$ as follows:

if $0 < \chi \leq 1$ (and then $\bar{\chi}=1$ ), then $X_1$ and $X_2$ are asymptotically dependent in extreme high values and $\chi$ is a measure for strength of dependence,
if $\chi = 0$ and $-1 \leq \bar{\chi} < 1$ , then $X_1$ and $X_2$ are asymptotically independent in extreme high values and $\bar{\chi}$ is a measure for strength of dependence. If $\bar{\chi} >0$ , there is a positive association: simultanueous extreme high values occur more frequently than under exact independence. If $\bar{\chi} <0$ , there is a negative association: simultanueous extreme high values occur less frequently than under exact independence.

Lower tail dependence coefficient

We denote by $\lambda_L$ the lower tail dependence coefficient:

$\lambda_L = \lim_{u \to 0} [F_2(X_2) < u| F_1(X_1) < u]$

provided that the limit exists.

The $\lambda_L$ coefficient can be interpreted as the tendency for one variable to take extreme low values given that the other variable is extremely low.

The variables $(X_1, X_2)$ are said to be:

asymptotically independent if and only if $\lambda_L=0$ ,
asymptotically dependent if and only if $0 < \lambda_L \leq 1$ .

Similarly to what is proposed for the upper tail coefficient, we can define the function $\chi_L(u)$ by:

$\chi_L(u) = \frac{\log (1 - C(u,u))}{\log (1-u)}, \forall u \in [0,1]$

We can see the tail dependence coefficient as the limit of the function $\chi(u)$ when $u$ tends to $0$ . As a matter of fact, when $u$ is close to $0$ , we have:

$\chi_L(u) = \frac{C(u,u)}{u} + o(1) = \Pset[F_2(X_2) < u | F_1(X_1) < u] + o(1)$

which proves that:

$\lim_{u \to 0} \chi_L(u) = \lambda_L$

We show that $0 \leq \chi_L(u) \leq 1$ .

Next to providing the limit $\lambda_L$ , the function $\chi_L(u)$ also provides some insight in the dependence structure of the variables at upper quantile levels: The larger $|\chi_L(u)|$ the more correlated are the variables.

The function $\chi_L(u)$ is estimated on data for several $u$ which must not be too low because of a problem of lack of data lesser than $u$ . Condifence intervals can be estimated on each $\chi_L(u)$ . If the confidence interval contains the zero value when $u$ tends to $0$ , then we can assume that $\lambda_L=0$ .

Within the class of asymptotically dependent variables, the value of $\chi_L$ increases with increasing degree of dependence at extreme levels.

We illustrate two cases where the variables are:

asymptotically independent: we generated a sample of size $10^3$ from a Frank copula which has a zero lower tail coefficient whatever the parameter,
asymptotically dependent: we generated a sample of size $10^3$ from a Clayton copula parametrized by $\theta = 2.0$ which has a positive lower tail coefficient.

(Source code, png)

Lower extremal dependence coefficient

Within the class of asymptotically independent variables, the degrees of relative strength of dependence is given by the function $\chi_L(u)$ defined by:

$\bar{\chi}_L(u) = \frac{\log (\Pset [F_1(X_1) < u] \Pset [F_2(X_2) < u])}{\log \Pset [F_1(X_1) < u, F_2(X_2) < u]} - 1, \forall u \in [0,1]$

We show that:

$\bar{\chi}_L(u) = \frac{2 \log u}{\log C(u,u)} - 1, \forall u \in [0,1]$

And we can define the lower extremal dependence coefficient by:

$\bar{\chi}_L = \lim_{u \to 0} \bar{\chi}_L(u)$

We show that $-1 \leq \bar{\chi}_L \leq 1$ and that if the variables are asymptotically dependent, then $\bar{\chi}_L=1$ :

$\lambda_L > 0 \Rightarrow \lim_{u \to 0} \bar{\chi}_L(u) = 1$

We illustrate the function $\bar{\chi}_L(u)$ for both previous cases: the Frank copula $\bar{\chi}(u)$ function is on the left and the Clayton copula $\bar{\chi}(u)$ function is on the right.

(Source code, png)

As a result, the pair $(\chi_L, \bar{\chi}_L)$ can be used as a summary of extremal dependence of $\vect{X} = (X_1, X_2)$ as follows:

if $0 < \chi_L \leq 1$ (and then $\bar{\chi}_L=1$ ), then $X_1$ and $X_2$ are asymptotically dependent in extreme low values and $\chi$ is a measure for strength of dependence,
if $\chi_L = 0$ and $-1 \leq \bar{\chi}_L < 1$ , then $X_1$ and $X_2$ are asymptotically independent in extreme low values and $\bar{\chi}$ is a measure for strength of dependence. If $\bar{\chi}_L >0$ , there is a positive association: simultaneous extreme low values occur more frequently than under exact independence. If $\bar{\chi}_L <0$ , there is a negative association: simultaneous extreme low values occur less frequently than under exact independence.

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