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62E20, Asymptotic distribution theory — 1 results found.

      
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On the Pickands Stochastic Process
C. R. Math. Rep. Acad. Sci. Canada Vol. 34 (2) 2012, pp. 39–49
Adja Mbarka Fall; Gane Samb Lo (Received: 2011/11/16)

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We consider the Pickands process \(P_{n}(s)
= \log (1/s)^{-1} \log \frac{X_{n-k+1,n}-X_{n-[k/s]+1,n}}{X_{n-[k/s]+1,n}-X_{n-[k/s^{2}]+1,n}},
\tag{1}
\qquad
\Bigl( \frac{k}{n} \leq s^2 \leq 1 \Bigr)\)
which is a generalization of the classical Pickands estimate \(P_{n}(1/2)\) of the extremal index. We undertake here a purely stochastic process view for the asymptotic theory of that process by using the Csörg–Csörg–Horvàth–Mason (1986). weighted approximation of the empirical and quantile processes to suitable Brownian bridges. This leads to the uniform convergence of the margins of this process to the extremal index and a complete theory of weak convergence of \(P_n\) in \(\ell^{\infty}([a,b])\) to some Gaussian process \[\left\{\mathbb{G},a\leq s \leq b\right\} \tag{2}\] for all \([a,b] \subset \left] 0,1 \right[\). This frame greatly simplifies the former results and enable applications based on stochastic processes methods.

Nous considérons le processus de Pickands défini en (1) qui est une généralisation de l’estimateur classique de Pickands \(P_{n}(1/2)\) de l’indice extremal. Nous abordons l’étude de ce processus du point de vue des processus stochastiques en établissant son comportement asymptotique. Nous utilisons comme outil principal l’approximation simultanée du processus empirique et du processus des quantiles uniformes dûe à Csörg–Csörg–Horvàth–Mason (1986). Nous établissons la convergence vague et uniforme du processus (1) vers un processus gaussien (2) entièrement décrit. Cette approche simplifie les résultats antérieurs et permet des applications basées sur des méthodes de processus stochastiques.

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