A regeneration-based runs estimator for the extremal index in the Markov setup

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

It is the purpose of this paper to introduce a novel estimator for the extremal index of an instantaneous function {f(Xn)}n of a regenerative Harris Markov chain X, based on the renewal properties of the latter. The estimate proposed may be viewed as a "regenerative version" of the runs estimator, insofar as it measures the clustering tendency of high threshold exceedances within regeneration cycles. Strong consistency of this estimator is established under mild stochastic stability assumptions and a simulation result is displayed in the case when the underlying chain is the waiting process related to a simple M/M/1 queue.

Mots clés: Regenerative Markov Chain; Extremal Index; Runs Estimator

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

Patrice Bertail, Stéphan Clémençon, Jessica Tressou. A regeneration-based runs estimator for the extremal index in the Markov setup. 2008. ⟨hal-00214305v2⟩