Extremal behaviour of stationary processes: the calibration technique in the extremal index estimation

D. Prata Gomes; Maria Manuela Neves

Discussiones Mathematicae Probability and Statistics (2010)

  • Volume: 30, Issue: 1, page 21-33
  • ISSN: 1509-9423

Abstract

top
Classical extreme value methods were derived when the underlying process is assumed to be a sequence of independent random variables. However when observations are taken along the time and/or the space the independence is an unrealistic assumption. A parameter that arises in this situation, characterizing the degree of local dependence in the extremes of a stationary series, is the extremal index, θ. In several areas such as hydrology, telecommunications, finance and environment, for example, the dependence between successive observations is observed so large values tend to occur in clusters. The extremal index is a quantity which, in an intuitive way, allows one to characterise the relationship between the dependence structure of the data and their extremal behaviour. Several estimators have been studied in the literature, but they endure a problem that usually appears in semiparametric estimators - a strong dependence on the high level uₙ, with an increasing bias and a decreasing variance as the threshold decreases. The calibration technique (Scheffé, 1973) is here considered as a procedure of controlling the bias of an estimator. It also leads to the construction of confidence intervals for the extremal index. A simulation study was performed for a stationary sequence and two sets of stationary data are under study for applying this technique.

How to cite

top

D. Prata Gomes, and Maria Manuela Neves. "Extremal behaviour of stationary processes: the calibration technique in the extremal index estimation." Discussiones Mathematicae Probability and Statistics 30.1 (2010): 21-33. <http://eudml.org/doc/277047>.

@article{D2010,
abstract = {Classical extreme value methods were derived when the underlying process is assumed to be a sequence of independent random variables. However when observations are taken along the time and/or the space the independence is an unrealistic assumption. A parameter that arises in this situation, characterizing the degree of local dependence in the extremes of a stationary series, is the extremal index, θ. In several areas such as hydrology, telecommunications, finance and environment, for example, the dependence between successive observations is observed so large values tend to occur in clusters. The extremal index is a quantity which, in an intuitive way, allows one to characterise the relationship between the dependence structure of the data and their extremal behaviour. Several estimators have been studied in the literature, but they endure a problem that usually appears in semiparametric estimators - a strong dependence on the high level uₙ, with an increasing bias and a decreasing variance as the threshold decreases. The calibration technique (Scheffé, 1973) is here considered as a procedure of controlling the bias of an estimator. It also leads to the construction of confidence intervals for the extremal index. A simulation study was performed for a stationary sequence and two sets of stationary data are under study for applying this technique.},
author = {D. Prata Gomes, Maria Manuela Neves},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {extreme value; stationary sequences; extremal index; estimation; calibration technique},
language = {eng},
number = {1},
pages = {21-33},
title = {Extremal behaviour of stationary processes: the calibration technique in the extremal index estimation},
url = {http://eudml.org/doc/277047},
volume = {30},
year = {2010},
}

TY - JOUR
AU - D. Prata Gomes
AU - Maria Manuela Neves
TI - Extremal behaviour of stationary processes: the calibration technique in the extremal index estimation
JO - Discussiones Mathematicae Probability and Statistics
PY - 2010
VL - 30
IS - 1
SP - 21
EP - 33
AB - Classical extreme value methods were derived when the underlying process is assumed to be a sequence of independent random variables. However when observations are taken along the time and/or the space the independence is an unrealistic assumption. A parameter that arises in this situation, characterizing the degree of local dependence in the extremes of a stationary series, is the extremal index, θ. In several areas such as hydrology, telecommunications, finance and environment, for example, the dependence between successive observations is observed so large values tend to occur in clusters. The extremal index is a quantity which, in an intuitive way, allows one to characterise the relationship between the dependence structure of the data and their extremal behaviour. Several estimators have been studied in the literature, but they endure a problem that usually appears in semiparametric estimators - a strong dependence on the high level uₙ, with an increasing bias and a decreasing variance as the threshold decreases. The calibration technique (Scheffé, 1973) is here considered as a procedure of controlling the bias of an estimator. It also leads to the construction of confidence intervals for the extremal index. A simulation study was performed for a stationary sequence and two sets of stationary data are under study for applying this technique.
LA - eng
KW - extreme value; stationary sequences; extremal index; estimation; calibration technique
UR - http://eudml.org/doc/277047
ER -

References

top
  1. [1] F. Andrews, Calibration and statistical inference, J. Ann. Statist. Assoc. 65 (1970), 1233-1242. Zbl0214.46803
  2. [2] S.G. Coles, J.A. Tawn and R.L. Smith, A sazonal Markov model for extremely low temperatures, Environmetrics 5 (1994), 221-339. 
  3. [3] P. Deheuvels, Point processes and multivariate extreme values, J. of multivariate analysis 13 (1983), 257-272. Zbl0519.60045
  4. [4] M.I. Gomes, Statistical inference in an extremal markovian model, COMPSTAT (1990), 257-262. 
  5. [5] M.I. Gomes, Modelos extremais em esquemas de dependência, I Congresso Ibero-Americano de Esdadistica e Investigacion Operativa (1992), 209-220. 
  6. [6] M.I. Gomes, On the estimation of parameters of rare events in environmental time series, Statistics for the Environment (1993), 226-241. 
  7. [7] T. Hsing, Extremal index estimation for weakly dependent stationary sequence, Ann. Statist 21 (1993), 2043-2071. Zbl0797.62018
  8. [8] M.R. Leadbetter, G. Lindgren and H. Rootzen, Extremes and related properties of random sequences and series, Springer Verlag. New York 1983. Zbl0518.60021
  9. [9] S. Nandagopalan, Multivariate Extremes and Estimation of the Extremal Index, Ph.D. Thesis. Techn. Report 315, Center for Stochastic Processes, Univ. North-Caroline 1990. Zbl0881.60050
  10. [10] D. Prata Gomes, Métodos computacionais na estimação pontual e intervalar do índice extremal. Tese de Doutoramento, Universidade Nova de Lisboa, Faculdade de Cięncias e Tecnologia 2008. 
  11. [11] H. Scheffé, A statistical theory of calibration, Ann. Statist 1 (1973), 1-37. Zbl0253.62023
  12. [12] R. Smith and I. Weissman, Estimating the extremal index, J. R. Statist. Soc. B, 56 (1994), 515-528. Zbl0796.62084
  13. [13] I. Weissman and S. Novak, On blocks and runs estimators of the extremal index, J. Statist. Plann. Inf. 66 (1998), 281-288. Zbl0953.62089
  14. [14] E.J. Williams, Regression methods in calibration problems, Proc. 37th Session, Bull. Int. Statist. Inst. 43 (1) (1969), 17-28. Zbl0214.46804

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.