An efficient semiparametric maxima estimator of the extremal index

The extremal index $\theta$, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate $\theta$ semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are competitive with the leading estimators. An application to sea-surge heights combines inferences about $\theta$ with a standard extreme value analysis of block maxima to estimate marginal quantiles.Comment: 17 pages, 7 figures. Minor edits made to version 1 prior to journal publication. The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-015-0221-

On extremal dependence : some contributions

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.Fundação para a Ciência e a Tecnologia (FCT