Workshop “On Some Current Research Topics in Extreme Value Theory”

Workshop Announcement

 On some current research topics in extreme value theory

(Organized by the FCT/MCTES  research project Spatial Extremes and CEAUL)

On December 13 we shall have a one-day meeting with contributions on current results and problems in extreme value theory.

 Program: 10h00-11h00 Michael Falk, University of Wuerzburg: On Extreme Value Processes and the Functional D-Norm

Abstract:

We introduce some mathematical framework from functional extreme value theory and provide basic definitions and tools. In particular we introduce a functional domain of attraction approach

for stochastic processes, which is more general than the usual one based on weak convergence.

The distribution function G of a continuous max-stable process on [0,1] is introduced and it is shown that G can be represented via a norm on functional space, called D-norm. This is in complete

accordance with the multivariate case and leads to the definition of functional generalized Pareto distributions W.

These satisfy W=1+log(G) in their upper tails, again in complete accordance with the uni- or multivariate case.

Applying  this framework to copula processes we derive characterizations of the domain of attraction condition for copula processes in terms of tail equivalence with a functional GPD.

delta-neighborhoods of a functional GPD are introduced and it is shown that these are characterized by a polynomial rate of convergence of functional extremes, which is well-known

in the multivariate case.

(by Stefan Aulbach, Michael Falk and Martin Hofmann)

11h00-12h00 Armelle Guillou, Université de Strasbourg: IRMA: Some Extensions of the Madogram

Abstract:

This talk is divided into two parts:

– the first one based on a paper published in Biometrika (2009) with Cooley, Diebolt and Naveau;

– the second  one, only in progress, based on some extension of the  madogram with Naveau and Schorgen.

In the first part, we model pairwise dependence of temporal maxima, such as annual maxima of precipitation, that have been recorded in space, either on a regular grid or at irregularly spaced

locations.

The construction of our estimators stems from the variogram concept.

The asymptotic properties of our pairwise dependence estimators are established through properties of empirical processes.

The performance of our approach is illustrated by simulations and by the treatment of a real dataset. However our assumption does not allow asymptotic independence.

To solve this issue,we propose to use an approach introduced by Ramos and Ledford (2009) (see also Ledford and Tawn, 1996 ) based on the introduction of a new parameter eta which measure

the dependence between the marginal tails.

This allows us to define a new eta-madogram from which we define a new extremal coefficient. Some simulations are provided in order to illustrate its behaviour.

 Lunch 14h00-15h00 Helena Ferreira, University of Beira Interior: The Multivariate Extremal Index and Tail Dependence

Abstract:

We compare the tail dependence parameters of the common distribution of a stationary sequence with the corresponding parameters in the limiting multivariate extreme

value model.

The multivariate extremal index is the key tool to quantify the effect of the dependence across the sequence in the tail dependence of the limiting MEV distribution of the

vector of componentwise maxima.

We illustrate the results with M4 processes.

15h00-15h30 Coffe-break

15h30-16h30 Pavlina Kalcheva Jordanova, Shumen University: Compound Extremal Processes and their Relation with Corresponding Sum and Point Processes.Abstract:

Limit theorems are widely used in practice. Here we consider point processes and associated with them processes of maxima and processes with additive increments.

The points have a time and a space component. In order to obtain limit theorems for their approximation we reduce the investigated models to so called accompanying

processes with non-random time points.

We transform the time and the state space properly in order to obtain sequences converging to a non-degenerate process.

The time space changes are regular and such that the number of time-points in any time interval [0, t] gets larger and the values of the state points gets smaller.

Finally we obtain different transfer theorems for sum and extremal processes and investigate the properties of the limiting processes.

Place: Anfiteatro da Fundação FCUL (C1, 3ºpiso), Faculdade de Ciências de Universidade de Lisboa.Free registration required: please confirm your presence to anafh@isa.utl.ptuntil December 4.  For any questions please contact anafh@isa.utl.pt (Organizer: Ana Ferreira)