# 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 (which is the notation of the probability community) as well as (which is the notation of the extreme value community),

• the upper extremal dependence coefficient denoted by ,

• the lower tail dependence coefficient denoted by ,

• the upper extremal dependence coefficient denoted by .

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

Let be a bivariate random vector with marginal distribution functions and , and copula .

Upper tail dependence coefficient

We denote by or the upper tail dependence coefficient:

provided that the limit exists.

The 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 are said to be:

• asymptotically independent if and only if ,

• asymptotically dependent if and only if .

Now, we define the function defined by:

We can see the tail dependence coefficient as the limit of the function when tends to . As a matter of fact, when is close to , we have:

which proves that:

Next to providing the limit , the function also provides some insight in the dependence structure of the variables at lower quantile levels: the larger the more correlated are the variables.

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

Within the class of asymptotically dependent variables, the value of 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 from a Frank copula which has a zero upper tail coefficient whatever the parameter,

• asymptotically dependent: we generated a sample of size from a Gumbel copula parametrized by which has a positive upper tail coefficient.

Upper extremal dependence coefficient

Within the class of asymptotically independent variables, the degrees of relative strength of dependence is given by the function defined by:

We show that:

where is the copula survivor function defined by:

And we can define the upper extremal dependence coefficient by:

We show that and that if the variables are asymptotically dependent, then :

We illustrate the function for both previous cases.

As a result, the pair can be used as a summary of extremal dependence of as follows:

• if (and then ), then and are asymptotically dependent in extreme high values and is a measure for strength of dependence,

• if and , then and are asymptotically independent in extreme high values and is a measure for strength of dependence. If , there is a positive association: simultanueous extreme high values occur more frequently than under exact independence. If , there is a negative association: simultanueous extreme high values occur less frequently than under exact independence.

Lower tail dependence coefficient

We denote by the lower tail dependence coefficient:

provided that the limit exists.

The 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 are said to be:

• asymptotically independent if and only if ,

• asymptotically dependent if and only if .

Similarly to what is proposed for the upper tail coefficient, we can define the function by:

We can see the tail dependence coefficient as the limit of the function when tends to . As a matter of fact, when is close to , we have:

which proves that:

We show that .

Next to providing the limit , the function also provides some insight in the dependence structure of the variables at upper quantile levels: The larger the more correlated are the variables.

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

Within the class of asymptotically dependent variables, the value of 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 from a Frank copula which has a zero lower tail coefficient whatever the parameter,

• asymptotically dependent: we generated a sample of size from a Clayton copula parametrized by which has a positive lower tail coefficient.

Lower extremal dependence coefficient

Within the class of asymptotically independent variables, the degrees of relative strength of dependence is given by the function defined by:

We show that:

And we can define the lower extremal dependence coefficient by:

We show that and that if the variables are asymptotically dependent, then :

We illustrate the function for both previous cases: the Frank copula function is on the left and the Clayton copula function is on the right.

As a result, the pair can be used as a summary of extremal dependence of as follows:

• if (and then ), then and are asymptotically dependent in extreme low values and is a measure for strength of dependence,

• if and , then and are asymptotically independent in extreme low values and is a measure for strength of dependence. If , there is a positive association: simultaneous extreme low values occur more frequently than under exact independence. If , there is a negative association: simultaneous extreme low values occur less frequently than under exact independence.