For stationary sequences, under general dependence restrictions, any limiting point process for time normalized upcrossings of high levels is a compound Poisson process, i.e., there is a clustering of high upcrossings, where the underlying Poisson points represent cluster positions and the multiplicities correspond to cluster sizes. For such classes of stationary sequences, there exists the upcrossings index η, 0≤η≤1, which is directly related to the extremal index θ, 0≤θ≤1, for suitable high levels. In this paper, we consider the problem of estimating the upcrossings index η for a class of stationary sequences satisfying a mild oscillation restriction. For the proposed estimator, properties such as consistency and asymptotic normality are studied. Finally, the performance of the estimator is assessed through simulation studies for autoregressive processes and case studies in the fields of environment and finance. Comparisons with other estimators derived from well known estimators of the extremal index are also presented.
KeywordsUpcrossings index Local dependence conditions Consistency and asymptotic normality
Mathematics Subject Classification (2000)60G70
We are grateful to the anonymous referees for their valuable criticisms, corrections and suggestions which helped considerably the final form of this paper.
We acknowledge the support from research unit “Centro de Matemática” of the University of Beira Interior, the research project PTDC/MAT/108575/2008 through the Foundation for Science and Technology (FCT) co-financed by FEDER/COMPETE and SFRH/BD/41439/2007.
- Drees H (2011) Bias correction for estimators of the extremal index. arXiv:1107.0935v1
- Ferreira H (2007) Runs of high values and the upcrossings index for a stationary sequence. In: Proceedings of the 56th session of the ISI Google Scholar
- Ferro CAT, Segers J (2002) Automatic declustering of extreme values via an estimator for the extremal index. Technical Report 2002-025. EURANDOM Eindhoven http://alexandria.tue.nl/repository/books/557750.pdf
- Laurini F, Tawn J (2006) The extremal index for GARCH(1, 1) processes with t-distributed innovations. http://www.scientificcommons.org/17368349
- Leadbetter MR, Nandagopalan S (1988) On exceedance point process for stationary sequences under mild oscillation restrictions. In: Hüsler J, Reiss D (eds) Extreme value theory: proceedings, Oberwolfach 1987. Springer, New York, pp 69–80 Google Scholar
- Liu RY, Singh K (1992) Moving blocks jackknife and bootstrap capture weak dependence. In: Lepage R, Billard L (eds) Exploring the limits of bootstrap. Wiley, New York, pp 225–248 Google Scholar
- Nandagopalan S (1990) Multivariate extremes and estimation of the extremal index. Ph.D. Thesis, University of North Carolina, Chapel Hill Google Scholar
- Politis DN, Romano JP (1992) A circular block resampling procedure for stationary data. In: Lepage R, Billard L (eds) Exploring the limits of bootstrap. Wiley, New York, pp 263–270 Google Scholar
- Singh K (1981) On the asymptotic accuracy of the Efron’s bootstrap. Ann Stat 9:345–362 Google Scholar