Regenerative Block-Bootstrap Confidence Intervals for the Extremal Index

Patrice Bertail (1) , Stéphan Clémençon (2) , Jessica Tressou (3)

1 MODAL'X - Modélisation aléatoire de Paris X
2 LTCI - Laboratoire Traitement et Communication de l'Information
3 MET@RISK - Méthodologies d'Analyse de Risque Alimentaire

This paper is devoted to show how the regenerative block-bootstrap methodology (RBB), proved asymptotically valid in the case of sample means and U-statistics based on data drawn from a regenerative arkov chain X = {Xn}n in Bertail and Clemencon (2005, 2006b), may be successfully extended for onstructing confidence intervals for the extremal index of instantaneous functions {f(Xn)}n. Precisely, this boils down to applying the RBB procedure to the regenerative blocks estimator proposed in Bertail et al. (2007) for measuring the clustering tendency of high threshold exceedances. Asymptotic normality of this estimator is established, together with the asymptotic validity of the bootstrap distribution estimate, under mild stochastic stability assumptions. Eventually, a preliminary numerical experiment is presented, with the aim to empirically evaluate the capacity of the RBB for building accurate confidence intervals for the extremal index of the waiting process related to a M/M/1 queue.

Mots clés: Atomic Markov chain; extremal index; regenerative block-bootstrap

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Patrice Bertail, Stéphan Clémençon, Jessica Tressou. Regenerative Block-Bootstrap Confidence Intervals for the Extremal Index. 2008. ⟨hal-00214306v2⟩