On extremal dependence: some contributions

TEST, 21(3) (2012), 566-583
http://dx.doi.org/10.1007/s11749-011-0261-3

The usual coefficients of tail dependence are based on exceedances of high values. These extremal events are useful and widely used in literature but an adverse situation may also occur with the upcrossing of a high level. In this context we define upcrossings-tail dependence coefficients and analyze all types of dependence coming out. We will prove that these coefficients are related to multivariate tail dependence coefficients already known in literature. We shall see that the upcrossings-tail dependence coefficients have the interesting feature of congregating both “temporal” and “spatial” dependence.

The coefficients of tail dependence can also be applied to stationary sequences and hence measure the tail dependence in time. Results concerning connections with the extremal index and the upcrossings index as well as with local dependence conditions will be stated. Several illustrative examples will be exploited and a small note on inference will be given by presenting estimators derived from the stated results and respective properties.