On the Pickands stochastic process
Adja Mbarka Fall, Gane Samb Lo
Applied Mathematics, Gaston Berger University, Saint-Louis, Senegal

We introduce a Pickands process Pn(s) as a generalization of the classical Pickands estimate Pn(1/2) of the extremal index. We undertake here a purely stochastic process view for the asymptotic theory of that process by using the Csörgö-Csörgö-Horvàth-Mason weighted empirical and quantile process to suitable brownian bridge. This leads to its uniform convergence to the extremal index is studied as well as a weak convergence in the l([a,b]) to some Gaussian process {G(s), a ≤s ≤b} for all [a,b] in ]0,1. Greatly simplify the former results and enable applications based on stochastic processes methods.

Bibliography:

Csörghö, M.,Csörghö, S., Horvàth, L. and Mason, M. (1986). Weighted empirical and quantile processes. Ann. Probab., 14, 31-85.

Drees. H. (1995). A refined Pickans Estimators for the extrem value index. Annals of Statistics Volume 23, Number 6, 2059-2080.

de Haan, L. and Feireira A. (2006). Extreme value theory: An introduction. Springer.

Lo, G. S.(1989). A note on the asymptotic normality of sums of extreme values. J. Statist. Plan. and Inf. 22, 89-94. (MR0996806)

Resnick, S.I. (1987). Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New-York.

Shorack G.R. and Wellner J. A.(1986). Empirical Processes with Applications to Statistics. Wiley-Interscience, New-York.

Keywords: Stochastic process; Regularly varying; Brownian motion; Asymptotic normality

Biography: I'm Adja Mbarka Fall, born November 23, 1984 in Dakar, Senegal. I did all my university studies at the University Gaston Berger of Saint-Louis in the field of applied sciences and technologies. I am currently a PhD in Applied Statistics.