Electronic Journal of Statistics

Regenerative block-bootstrap confidence intervals for tail and extremal indexes

Patrice Bertail, Stéphan Clémençon, and Jessica Tressou

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1366896904

Digital Object Identifier
doi:10.1214/13-EJS807

Mathematical Reviews number (MathSciNet)
MR3056073

Zentralblatt MATH identifier
1329.60146

Subjects
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]

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

Citation

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.


Export citation

References

  • [1] Ancona-Navarette, M. A. and Tawn, J. A. (2000). A Comparison of Methods for Estimating the Extremal Index., Extremes 3 5–38.
  • [2] Asmussen, S. (1998). Extreme Value Theory for Queues Via Cycle Maxima., Extremes 1 137–168.
  • [3] Asmussen, S. (2003)., Applied Probability and Queues. Springer-Verlag, New York.
  • [4] Athreya, K. B. and Pantula, S. G. (1986). Mixing Properties of Harris Chains and Autoregressive Processes., Journal of Applied Probability 23 880–892.
  • [5] Bacro, J. N. and Brito, M. (1998). A tail bootstrap procedure for estimating the tail Pareto-index., Journal of Statistical Planning and Inference 71 245–260.
  • [6] Beirlant, J., Dierckx, G., Goegebeur, Y. and Matthys, G. (1999). Tail index estimation and an exponential regression model., Extremes 2 177–200.
  • [7] Bertail, P. and Clémençon, S. (2006a). Regeneration-based statistics for Harris recurrent Markov chains. In, Dependence in Probability and Statistics ( P. Bertail, P. Soulier and P. Doukhan, eds.). Lecture Notes in Statistics 187 3–54.
  • [8] Bertail, P. and Clémençon, S. (2006b). Regenerative-Block Bootstrap for Markov Chains., Bernoulli 12 689–712.
  • [9] Bertail, P. and Clémençon, S. (2007). Approximate regenerative block-bootstrap for Markov chains., Computational Statistics and Data Analysis 52 2739–2756.
  • [10] Bertail, P., Clémençon, S. and Tressou, J. (2009). Extreme value statistics for Markov chains via the (pseudo-)regenerative method., Extremes 12 327–360.
  • [11] Coles, S. (2001)., An introduction to statistical modelling of Extreme Values. Springer series in Statistics. Springer.
  • [12] Danielsson, J., de Haan, L., Peng, L. andde Vries, C. G. (2001). Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation., Journal of Multivariate Analysis 76 226–248.
  • [13] de Haan, L. and Peng, L. (1998). Comparison of tail index estimators., Statistica Neerlandica 52 60–70.
  • [14] Politis, D. N., Romano, J. P. and Wolf, M. (2001). On the asymptotic theory of subsampling., Statistica Sinica 114 1105–1124.
  • [15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997)., Modelling Extremal Events for Insurance and Finance. Applications of Mathematics. Springer-Verlag.
  • [16] Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extreme values., Journal of the Royal Statistical Society 65 545–556.
  • [17] Feuerverger, A. and Hall, P. (1999). Estimating a tail exponent by modelling departure from a Pareto Distribution., Annals of Statistics 27 760–781.
  • [18] Finkenstadt, B. and Rootzén, H. (2003)., Extreme values in Finance, Telecommunications and the Environment. Monograph on Statistics and Applied Probability 99. Chapman & Hall.
  • [19] Goldie, C. M. and Smith, R. L. (1987). Slow variation with remainder: theory and applications., Quarterly Journal of Mathematics 38 45–71.
  • [20] Hall, P. (1990). Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems., Journal of Multivariate Analysis 32 177–203.
  • [21] Hsing, T. (1993). Extremal index estimation for a weakly dependent stationary sequence., Annals of Statistics 21 2043–2071.
  • [22] Jain, J. and Jamison, B. (1967). Contributions to Doeblin’s theory of Markov processes., Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 8 19–40.
  • [23] Laurini, F. and Tawn, J. A. (2003). New Estimators for the Extremal Index and Other Cluster Characteristics., Extremes 6 189–211.
  • [24] Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences., Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 65 291–306.
  • [25] Meyn, S. P. and Tweedie, R. L. (1996)., Markov Chains and Stochastic Stability. Springer-Verlag.
  • [26] Nummelin, E. (1978). A splitting technique for Harris recurrent chains., Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 43 309–318.
  • [27] Resnick, S. and Stărică, C. (1998). Tail Index estimation for dependent data., Annals of Applied Probability 8 1156–1183.
  • [28] Revuz, D. (1984)., Markov Chains. 2nd edition, North-Holland.
  • [29] Robert, C. Y. (2009). Inference for the limiting cluster size distribution of extreme values., Annals of Statistics 37 271–310.
  • [30] Robert, C. Y., Segers, J. and Ferro, C. A. T. (2009). A sliding blocks estimator for the extremal index., Electronic Journal of Statistics 3 993-1020.
  • [31] Roberts, G. O., Rosenthal, J. S., Segers, J. and Sousa, B. (2006). Extremal indices, geometric ergodicity of Markov chains, and MCMC., Extremes 9 213–229.
  • [32] Rootzén, H. (1988). Maxima and exceedances of stationary Markov chains., Advances in Applied Probability 20 371–390.
  • [33] Weissman, I. and Novak, S. Y. (1998). On blocks and runs estimators of the extremal index., Journal of Statistical Planning and Inference 66 281–288.