Regenerative Block-Bootstrap Confidence Intervals for Tail and Extremal Indexes

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

A theoretically sound bootstrap procedure is proposed for building accurate confidence intervals of parameters describing the extremal behavior of instantaneous functionals of a Harris Markov chain, namely the extremal and tail indexes. Regenerative properties of the chain (or of a Nummelin extension of the latter) are here exploited in order to construct consistent estimators of these parameters, following the approach developed in Bertail, Clémençon & Tressou (Extremes, 2009). Their asymptotic normality is first established and the standardization problem is also tackled. It is then proved that, based on these estimators, the (approximate) regenerative block-bootstrap introduced in Bertail &Clémençon (2006) yields asymptotically valid confidence intervals. In order to illustrate the performance of the methodology studied in this paper, simulation results are additionally displayed.

Mots clés: Regenerative Markov chain; Nummelin splitting technique; Extreme value statistics; Cycle submaximum; Hill estimator; Extremal index; Regenerative-block bootstrap

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

Patrice Bertail, Stéphan Clémençon, Jessica Tressou. Regenerative Block-Bootstrap Confidence Intervals for Tail and Extremal Indexes. 2012. ⟨hal-00733139v2⟩