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 {f(X-n)}(n is an element of N) of a Harris Markov chain X, namely the extremal and tail indexes. Regenerative properties of the chain X (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 [10]. Their asymptotic normality is first established and the standardization problem is also tackled. It is then proved that, based on these estimators, there generative block-bootstrap and its approximate version, both introduced in [7], yield asymptotically valid confidence intervals. In order to illustrate the performance of the methodology studied in this paper, simulation results are additionally displayed.

Mots clés: MARKOV-CHAINS; regenerative-block bootstrap; extremal index; Regenerative Markov chain; Nummelin splitting technique; extreme value statistics; cycle submaximum; Hill estimator; ESTIMATORS; INFERENCE; VALUES

Domaines: Sciences agricoles

Patrice Bertail, Stéphan Clémençon, Jessica Tressou. Regenerative block-bootstrap confidence intervals for tail and extremal indexes. Electronic Journal of Statistics , 2013, 7, pp.1224 - 1248. ⟨10.1214/13-EJS807⟩. ⟨hal-01019098⟩