Annals of Statistics

Weak convergence of a pseudo maximum likelihood estimator for the extremal index

Betina Berghaus and Axel Bücher

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both disjoint and sliding blocks estimator for the extremal index are analyzed in detail. In contrast to many competitors, the estimators only depend on the choice of one parameter sequence. We derive an asymptotic expansion, prove asymptotic normality and show consistency of an estimator for the asymptotic variance. Explicit calculations in certain models and a finite-sample Monte Carlo simulation study reveal that the sliding blocks estimator outperforms other blocks estimators, and that it is competitive to runs- and inter-exceedance estimators in various models. The methods are applied to a variety of financial time series.

Article information

Source
Ann. Statist., Volume 46, Number 5 (2018), 2307-2335.

Dates
Received: August 2016
Revised: July 2017
First available in Project Euclid: 17 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1534492837

Digital Object Identifier
doi:10.1214/17-AOS1621

Mathematical Reviews number (MathSciNet)
MR3845019

Zentralblatt MATH identifier
06964334

Subjects
Primary: 62G32: Statistics of extreme values; tail inference 62E20: Asymptotic distribution theory 62M09: Non-Markovian processes: estimation
Secondary: 60G70: Extreme value theory; extremal processes 62G20: Asymptotic properties

Keywords
Clusters of extremes extremal index stationary time series mixing coefficients block maxima

Citation

Berghaus, Betina; Bücher, Axel. Weak convergence of a pseudo maximum likelihood estimator for the extremal index. Ann. Statist. 46 (2018), no. 5, 2307--2335. doi:10.1214/17-AOS1621. https://projecteuclid.org/euclid.aos/1534492837


Export citation

References

  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley, Chichester.
    Mathematical Reviews (MathSciNet): MR2108013
    Zentralblatt MATH: 1070.62036
  • Berghaus, B. and Bücher, A. (2018). Supplement to “Weak convergence of a pseudo maximum likelihood estimator for the extremal index.” DOI:10.1214/17-AOS1621SUPP.
  • Billingsley, P. (1979). Probability and Measure. Wiley, New York.
    Zentralblatt MATH: 0411.60001
  • Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107–144.
    Zentralblatt MATH: 1189.60077
    Digital Object Identifier: doi:10.1214/154957805100000104
  • Bücher, A. and Segers, J. (2015). Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series. Available at arXiv:1511.07613.
    arXiv: 1511.07613
  • Darsow, W. F., Nguyen, B. and Olsen, E. T. (1992). Copulas and Markov processes. Illinois J. Math. 36 600–642.
    Zentralblatt MATH: 0770.60019
    Project Euclid: euclid.ijm/1255987328
  • de Haan, L., Resnick, S. I., Rootzén, H. and de Vries, C. G. (1989). Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Process. Appl. 32 213–224.
    Zentralblatt MATH: 0679.60029
    Digital Object Identifier: doi:10.1016/0304-4149(89)90076-8
  • Drees, H. (2000). Weighted approximations of tail processes for $\beta$-mixing random variables. Ann. Appl. Probab. 10 1274–1301.
    Zentralblatt MATH: 1073.60520
    Project Euclid: euclid.aoap/1019487617
  • Drees, H. (2002). Tail empirical processes under mixing conditions. In Empirical Process Techniques for Dependent Data 325–342. Birkhäuser, Boston, MA.
    Zentralblatt MATH: 1021.62038
  • Drees, H. and Rootzén, H. (2010). Limit theorems for empirical processes of cluster functionals. Ann. Statist. 38 2145–2186.
    Zentralblatt MATH: 1210.62051
    Digital Object Identifier: doi:10.1214/09-AOS788
    Project Euclid: euclid.aos/1278861245
  • Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extreme values. J. R. Stat. Soc. Ser. B. Stat. Methodol. 65 545–556.
    Zentralblatt MATH: 1065.62091
    Digital Object Identifier: doi:10.1111/1467-9868.00401
  • Hsing, T. (1984). Point processes associated with extreme value theory. Ph.D. thesis, Univ. North Carolina at Chapel Hill, ProQuest LLC, Ann Arbor, MI.
  • Hsing, T. (1993). Extremal index estimation for a weakly dependent stationary sequence. Ann. Statist. 21 2043–2071.
    Zentralblatt MATH: 0797.62018
    Digital Object Identifier: doi:10.1214/aos/1176349409
    Project Euclid: euclid.aos/1176349409
  • Hsing, T., Hüsler, J. and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Probab. Theory Related Fields 78 97–112.
    Zentralblatt MATH: 0619.60054
    Digital Object Identifier: doi:10.1007/BF00718038
  • Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
    Zentralblatt MATH: 0291.60029
    Digital Object Identifier: doi:10.1007/BF02392040
    Project Euclid: euclid.acta/1485889791
  • Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65 291–306.
    Zentralblatt MATH: 0506.60030
    Digital Object Identifier: doi:10.1007/BF00532484
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Leadbetter, M. R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Probab. 16 431–478.
    Zentralblatt MATH: 0648.60039
    Digital Object Identifier: doi:10.1214/aop/1176991767
    Project Euclid: euclid.aop/1176991767
  • Northrop, P. J. (2015). An efficient semiparametric maxima estimator of the extremal index. Extremes 18 585–603.
    Zentralblatt MATH: 1327.62325
    Digital Object Identifier: doi:10.1007/s10687-015-0221-5
  • O’Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Probab. 15 281–291.
    Zentralblatt MATH: 0619.60025
    Digital Object Identifier: doi:10.1214/aop/1176992270
    Project Euclid: euclid.aop/1176992270
  • Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Probab. 4 529–548.
    Zentralblatt MATH: 0806.60041
    Digital Object Identifier: doi:10.1214/aoap/1177005071
    Project Euclid: euclid.aoap/1177005071
  • Rémillard, B., Papageorgiou, N. and Soustra, F. (2012). Copula-based semiparametric models for multivariate time series. J. Multivariate Anal. 110 30–42.
    Zentralblatt MATH: 1281.62136
    Digital Object Identifier: doi:10.1016/j.jmva.2012.03.001
  • Robert, C. Y. (2009). Inference for the limiting cluster size distribution of extreme values. Ann. Statist. 37 271–310.
    Zentralblatt MATH: 1158.62061
    Digital Object Identifier: doi:10.1214/07-AOS551
    Project Euclid: euclid.aos/1232115935
  • Robert, C. Y., Segers, J. and Ferro, C. A. T. (2009). A sliding blocks estimator for the extremal index. Electron. J. Stat. 3 993–1020.
    Zentralblatt MATH: 1326.60075
    Digital Object Identifier: doi:10.1214/08-EJS345
  • Rootzén, H. (2009). Weak convergence of the tail empirical process for dependent sequences. Stochastic Process. Appl. 119 468–490.
  • Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. Roy. Statist. Soc. Ser. B 56 515–528.
    Zentralblatt MATH: 0796.62084
  • Süveges, M. (2007). Likelihood estimation of the extremal index. Extremes 10 41–55.
  • Süveges, M., Davison, A. C. et al. (2010). Model misspecification in peaks over threshold analysis. Ann. Appl. Stat. 4 203–221.
    Zentralblatt MATH: 1189.62086
    Digital Object Identifier: doi:10.1214/09-AOAS292
    Project Euclid: euclid.aoas/1273584453
  • Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.
    Zentralblatt MATH: 0417.60073
    Digital Object Identifier: doi:10.2307/1426858
  • Weissman, I. and Novak, S. Y. (1998). On blocks and runs estimators of the extremal index. J. Statist. Plann. Inference 66 281–288.
    Zentralblatt MATH: 0953.62089
    Digital Object Identifier: doi:10.1016/S0378-3758(97)00095-5

Supplemental materials

  • Supplement to: “Weak convergence of a pseudo maximum likelihood estimator for the extremal index”. The supplement contains missing proofs for the results in this paper and additional simulation results.
    Digital Object Identifier: doi:10.1214/17-AOS1621SUPP
    Supplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.

The Institute of Mathematical Statistics

Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more