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2020 | OriginalPaper | Buchkapitel

A Note on Robust Estimation of the Extremal Index

verfasst von : M. Ivette Gomes, Miranda Cristina, Manuela Souto de Miranda

Erschienen in: Nonparametric Statistics

Verlag: Springer International Publishing

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Abstract

Many examples in the most diverse fields of application show the need for statistical methods of analysis of extremes of dependent data. A crucial issue that appears when there is dependency is the reliable estimation of the extremal index (EI), a parameter related to the clustering of large events. The most popular EI-estimators, like the blocks’ EI-estimators, are very sensitive to anomalous cluster sizes and exhibit a high bias. The need for robust versions of such EI-estimators is the main topic under discussion in this paper.

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Literatur
1.
Zurück zum Zitat Aitkin, M.: A general maximum likelihood analysis of overdispersion in generalized linear models. Stat. Comput. 6, 251–262 (1996)CrossRef Aitkin, M.: A general maximum likelihood analysis of overdispersion in generalized linear models. Stat. Comput. 6, 251–262 (1996)CrossRef
2.
Zurück zum Zitat Beath, K.: A mixture-based approach to robust analysis of generalised linear models. Journal of Applied Statist. 45, 2256–2268 (2017)MathSciNetCrossRef Beath, K.: A mixture-based approach to robust analysis of generalised linear models. Journal of Applied Statist. 45, 2256–2268 (2017)MathSciNetCrossRef
4.
Zurück zum Zitat Barry, S., Welsh, A.: Generalized additive modelling and zero inflated count data. Ecological Modelling 157, 179–188 (2002)CrossRef Barry, S., Welsh, A.: Generalized additive modelling and zero inflated count data. Ecological Modelling 157, 179–188 (2002)CrossRef
5.
6.
Zurück zum Zitat Bianco, A., Ben, M., Yohai, V.: Robust estimation for linear regression with asymmetric errors. The Canadian Journal of Statistics 33, 511–528 (2005)MathSciNetCrossRef Bianco, A., Ben, M., Yohai, V.: Robust estimation for linear regression with asymmetric errors. The Canadian Journal of Statistics 33, 511–528 (2005)MathSciNetCrossRef
7.
Zurück zum Zitat Cantoni, E., Ronchetti, E.: Robust inference for generalized linear models. J. Amer. Statist. Assoc. 96, 1022–1030 (2001)MathSciNetCrossRef Cantoni, E., Ronchetti, E.: Robust inference for generalized linear models. J. Amer. Statist. Assoc. 96, 1022–1030 (2001)MathSciNetCrossRef
8.
Zurück zum Zitat Cantoni, E., Ronchetti, E.: A robust approach for skewed and heavy-tailed outcomes in the analysis of health care expenditures. J. of Health Economics 25, 198–213 (2006)CrossRef Cantoni, E., Ronchetti, E.: A robust approach for skewed and heavy-tailed outcomes in the analysis of health care expenditures. J. of Health Economics 25, 198–213 (2006)CrossRef
9.
Zurück zum Zitat Cantoni, E., Zedini, A.: A robust version of the hurdle model. J. Statist. Plann. and Infer. 141, 1214–1223 (2011)MathSciNetCrossRef Cantoni, E., Zedini, A.: A robust version of the hurdle model. J. Statist. Plann. and Infer. 141, 1214–1223 (2011)MathSciNetCrossRef
10.
Zurück zum Zitat Dell’Aquila, R., Embrechts, P.: Extremes and robustness: a contradiction? Financial Markets and Portfolio Management 20, 103–118 (2006)CrossRef Dell’Aquila, R., Embrechts, P.: Extremes and robustness: a contradiction? Financial Markets and Portfolio Management 20, 103–118 (2006)CrossRef
11.
Zurück zum Zitat Ferreira, M.: Heuristic tools for the estimation of the extremal index: a comparison of methods. Revstat—Statist. J. 16, 115-13 (2018) Ferreira, M.: Heuristic tools for the estimation of the extremal index: a comparison of methods. Revstat—Statist. J. 16, 115-13 (2018)
12.
Zurück zum Zitat Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. Royal Statist. Soc., Series B 65, 545-556 (2003) Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. Royal Statist. Soc., Series B 65, 545-556 (2003)
13.
Zurück zum Zitat Gilleland, E., Katz, R.: extRemes 2.0: An extreme value analysis package in R. J. of Statistical Software 72, 1-39 (2016) Gilleland, E., Katz, R.: extRemes 2.0: An extreme value analysis package in R. J. of Statistical Software 72, 1-39 (2016)
14.
Zurück zum Zitat Gomes, M.I., Guillou, A.: Extreme value theory and statistics of univariate extremes: a review. International Statistical Review 83(2), 263–292 (2015)MathSciNetCrossRef Gomes, M.I., Guillou, A.: Extreme value theory and statistics of univariate extremes: a review. International Statistical Review 83(2), 263–292 (2015)MathSciNetCrossRef
15.
Zurück zum Zitat Gomes, M.I., Hall, A., Miranda, C.: Subsampling techniques and the Jackknife methodology in the estimation of the extremal index. Comput. Statist. & Data Anal. 52, 2022–2041 (2008)MathSciNetCrossRef Gomes, M.I., Hall, A., Miranda, C.: Subsampling techniques and the Jackknife methodology in the estimation of the extremal index. Comput. Statist. & Data Anal. 52, 2022–2041 (2008)MathSciNetCrossRef
16.
Zurück zum Zitat Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics: The Approach based on Influence Functions. John Wiley, New York (1986) Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics: The Approach based on Influence Functions. John Wiley, New York (1986)
17.
Zurück zum Zitat Heritier, S., Cantoni, E., Copt, S., Victoria-Feser, M.-P.: Robust Methods in Biostatistics. John Wiley & Sons, United Kingdom (2009) Heritier, S., Cantoni, E., Copt, S., Victoria-Feser, M.-P.: Robust Methods in Biostatistics. John Wiley & Sons, United Kingdom (2009)
18.
Zurück zum Zitat Huber, P.: Robust Statistics. John Wiley, New York (1981) Huber, P.: Robust Statistics. John Wiley, New York (1981)
20.
Zurück zum Zitat Hsing, T.: Extremal index estimation for a weakly dependent stationary sequence. The Ann. of Statist. 21, 2043–2071 (1993)MathSciNetCrossRef Hsing, T.: Extremal index estimation for a weakly dependent stationary sequence. The Ann. of Statist. 21, 2043–2071 (1993)MathSciNetCrossRef
21.
Zurück zum Zitat Hsing, T., Hüsler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Probab. Th. and Rel. Fields 78, 97–112 (1988)MathSciNetCrossRef Hsing, T., Hüsler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Probab. Th. and Rel. Fields 78, 97–112 (1988)MathSciNetCrossRef
22.
Zurück zum Zitat Laurini, F., Tawn, J.: New estimators for the extremal index and other cluster characteristics. Extremes 6, 189–211 (2003)MathSciNetCrossRef Laurini, F., Tawn, J.: New estimators for the extremal index and other cluster characteristics. Extremes 6, 189–211 (2003)MathSciNetCrossRef
23.
Zurück zum Zitat Leadbetter, M.R., Nandagopalan, S.: On exceedance point processes for stationary sequences under mild oscillation restrictions. In Hüsler, J. and R.-D. Reiss (eds.), Extreme Value Theory, Springer-Verlag, pp. 69–80 (1989) Leadbetter, M.R., Nandagopalan, S.: On exceedance point processes for stationary sequences under mild oscillation restrictions. In Hüsler, J. and R.-D. Reiss (eds.), Extreme Value Theory, Springer-Verlag, pp. 69–80 (1989)
24.
Zurück zum Zitat Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Series. Springer Verlag, New York (1983) Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Series. Springer Verlag, New York (1983)
27.
Zurück zum Zitat Min, Y., Agresti, A.: Modeling nonnegative data with clumping at zero: a survey. J. of the Iranian Statistical Society 1, 7–33 (2002)MATH Min, Y., Agresti, A.: Modeling nonnegative data with clumping at zero: a survey. J. of the Iranian Statistical Society 1, 7–33 (2002)MATH
28.
Zurück zum Zitat Northrop, P.: An efficient semiparametric maxima estimator of the extremal index. Extremes 18, 585–603 (2015)MathSciNetCrossRef Northrop, P.: An efficient semiparametric maxima estimator of the extremal index. Extremes 18, 585–603 (2015)MathSciNetCrossRef
30.
Zurück zum Zitat R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria (2014) R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria (2014)
31.
Zurück zum Zitat Robert, C.Y.: Inference for the limiting cluster size distribution of extreme values. The Ann. of Statist. 37, 271–310 (2009)MathSciNetCrossRef Robert, C.Y.: Inference for the limiting cluster size distribution of extreme values. The Ann. of Statist. 37, 271–310 (2009)MathSciNetCrossRef
32.
Zurück zum Zitat Robinson, M., Tawn, J.: Extremal analysis of processes sampled at different frequencies. J. R. Statist. Soc. B. 62, 117–135 (2000)MathSciNetCrossRef Robinson, M., Tawn, J.: Extremal analysis of processes sampled at different frequencies. J. R. Statist. Soc. B. 62, 117–135 (2000)MathSciNetCrossRef
33.
Zurück zum Zitat Smith, R.L., Weissman, I.: Estimating the extremal index. J. Royal Statist. Soc., Series B 56, 41–55 (1994) Smith, R.L., Weissman, I.: Estimating the extremal index. J. Royal Statist. Soc., Series B 56, 41–55 (1994)
35.
Zurück zum Zitat Süveges, M., Davison, A.: Model misspecification in peaks over threshold analysis. The Annals of Applied Statistics 4, 203–221 (2010)MathSciNetCrossRef Süveges, M., Davison, A.: Model misspecification in peaks over threshold analysis. The Annals of Applied Statistics 4, 203–221 (2010)MathSciNetCrossRef
36.
Zurück zum Zitat Valdora, M., Yohai, V.: Robust estimators for generalized linear models. J. Statist. Plann. and Infer. 146, 31–48 (2014)MathSciNetCrossRef Valdora, M., Yohai, V.: Robust estimators for generalized linear models. J. Statist. Plann. and Infer. 146, 31–48 (2014)MathSciNetCrossRef
37.
Zurück zum Zitat Vandewalle, B., Beirlant, J., Christmann, A., Hubert, M.: A robust estimator for the tail index of Pareto-type distributions. Comput. Statist. & Data Anal. 51, 6252–6268 (2007)MathSciNetCrossRef Vandewalle, B., Beirlant, J., Christmann, A., Hubert, M.: A robust estimator for the tail index of Pareto-type distributions. Comput. Statist. & Data Anal. 51, 6252–6268 (2007)MathSciNetCrossRef
Metadaten
Titel
A Note on Robust Estimation of the Extremal Index
verfasst von
M. Ivette Gomes
Miranda Cristina
Manuela Souto de Miranda
Copyright-Jahr
2020
DOI
https://doi.org/10.1007/978-3-030-57306-5_20

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