Electronic Journal of Statistics

Regenerative block-bootstrap confidence intervals for tail and extremal indexes

Patrice Bertail, Stéphan Clémençon, and Jessica Tressou

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A theoretically sound bootstrap procedure is proposed for building accurate confidence intervals of parameters describing the extremal behavior of instantaneous functionals $\{f(X_{n})\}_{n\in\mathbb{N}}$ 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, the regenerative 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.

Article information

Electron. J. Statist., Volume 7 (2013), 1224-1248.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]

Regenerative Markov chain Nummelin splitting technique extreme value statistics cycle submaximum Hill estimator extremal index regenerative-block bootstrap


Bertail, Patrice; Clémençon, Stéphan; Tressou, Jessica. Regenerative block-bootstrap confidence intervals for tail and extremal indexes. Electron. J. Statist. 7 (2013), 1224--1248. doi:10.1214/13-EJS807. https://projecteuclid.org/euclid.ejs/1366896904

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