The Annals of Statistics
- Ann. Statist.
- Volume 37, Number 1 (2009), 271-310.
Inference for the limiting cluster size distribution of extreme values
Full-text: Open access
Abstract
Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data.
Article information
Source
Ann. Statist. Volume 37, Number 1 (2009), 271-310.
Dates
First available in Project Euclid: 16 January 2009
Permanent link to this document
http://projecteuclid.org/euclid.aos/1232115935
Digital Object Identifier
doi:10.1214/07-AOS551
Mathematical Reviews number (MathSciNet)
MR2488352
Zentralblatt MATH identifier
1158.62061
Subjects
Primary: 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory 62M09: Non-Markovian processes: estimation
Secondary: 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference
Keywords
Extreme values exceedance point processes limiting cluster size distribution extremal index strictly stationary sequences
Citation
Robert, Christian Y. Inference for the limiting cluster size distribution of extreme values. Ann. Statist. 37 (2009), no. 1, 271--310. doi:10.1214/07-AOS551. http://projecteuclid.org/euclid.aos/1232115935.
References
- [1] Ancona-Navarrete, M. and Tawn, J. A. (2000). A comparison of methods for estimating the extremal index. Extremes 3 5–38.Mathematical Reviews (MathSciNet): MR1815172
Digital Object Identifier: doi: 10.1023/A:1009993419559 - [2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Mathematical Reviews (MathSciNet): MR233396
- [3] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.Mathematical Reviews (MathSciNet): MR1700749
- [4] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.Mathematical Reviews (MathSciNet): MR1324786
- [5] Buchmann, B. and Grübel, R. (2003). Decompounding: An estimation problem for Poisson random sums. Ann. Statist. 31 1054–1074.
- [6] Buchmann, B. and Grübel, R. (2004). Decompounding Poisson random sums: Recursively truncated estimates in the discrete case. Ann. Inst. Statist. Math. 56 743–756.
- [7] Deo, C. M. (1973). A note on empirical processes of strong-mixing sequences. Ann. Probab. 5 870–875.
- [8] Doukhan, P. (1994). Mixing. Properties and Examples. Springer, New York.Mathematical Reviews (MathSciNet): MR1312160
- [9] Drees, H. (1998). On smooth statistical tail functionals. Scand. J. Statist. 25 187–210.Mathematical Reviews (MathSciNet): MR1614276
Digital Object Identifier: doi: 10.1111/1467-9469.00097 - [10] Drees, H. (2000). Weighted approximations of tail processes for β-mixing random variables. Ann. Appl. Probab. 10 1274–1301.Mathematical Reviews (MathSciNet): MR1810875
Zentralblatt MATH: 1073.60520
Project Euclid: euclid.aoap/1019487617 - [11] Drees, H. (2002). Tail empirical processes under mixing conditions. In Empirical Process Techniques for Dependent Data (H. G. Dehling, T. Mikosch and M. Sorensen, eds.) 325–342. Birkhäuser, Boston.
- [12] Drees, H. (2003). Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9 617–657.Mathematical Reviews (MathSciNet): MR1996273
Digital Object Identifier: doi: 10.3150/bj/1066223272
Project Euclid: euclid.bj/1066223272 - [13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
- [14] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1008.Mathematical Reviews (MathSciNet): MR666121
Digital Object Identifier: doi: 10.2307/1912773
JSTOR: links.jstor.org - [15] Ferro, C. A. T. (2003). Statistical methods for clusters of extreme values. Ph.D. thesis, Lancaster Univ.
- [16] Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 65 545–556.Mathematical Reviews (MathSciNet): MR1983763
Zentralblatt MATH: 1065.62091
Digital Object Identifier: doi: 10.1111/1467-9868.00401
JSTOR: links.jstor.org - [17] Goldie, C. M. (1989). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.Mathematical Reviews (MathSciNet): MR1097468
Zentralblatt MATH: 0724.60076
Digital Object Identifier: doi: 10.1214/aoap/1177005985
Project Euclid: euclid.aoap/1177005985 - [18] de Haan, L. and Resnick, S. (1993). Estimating the limiting distribution of multivariate extremes. Commun. Statist. Stochatic Models 9 275–309.Mathematical Reviews (MathSciNet): MR1213072
Zentralblatt MATH: 0777.62036
Digital Object Identifier: doi: 10.1080/15326349308807267 - [19] de Haan, L., Resnick, S., Rootzen, H. and de Vries, C. G. (1989). Extremal behaviour of solutions to a stochastic difference equation with application to ARCH processes. Stochastic Process. Appl. 32 213–224.Mathematical Reviews (MathSciNet): MR1014450
Zentralblatt MATH: 0679.60029
Digital Object Identifier: doi: 10.1016/0304-4149(89)90076-8 - [20] Hsing, T. (1984). Point processes associated with extreme value theory. Ph.D. thesis, Dept. Statistics, Univ. North Carolina.
- [21] Hsing, T. (1991). Estimating the parameters of rare events. Stochastic Process. Appl. 37 117–139.Mathematical Reviews (MathSciNet): MR1091698
Zentralblatt MATH: 0722.62021
Digital Object Identifier: doi: 10.1016/0304-4149(91)90064-J - [22] Hsing, T. (1993). Extremal index estimation for a weakly dependent stationary sequence. Ann. Statist. 21 2043–2071.Mathematical Reviews (MathSciNet): MR1245780
Zentralblatt MATH: 0797.62018
Digital Object Identifier: doi: 10.1214/aos/1176349409
Project Euclid: euclid.aos/1176349409 - [23] Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Probab. Theory Related Fields 95 331–356.Mathematical Reviews (MathSciNet): MR1213195
Zentralblatt MATH: 0791.60024
Digital Object Identifier: doi: 10.1007/BF01192168 - [24] 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.
- [25] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.Mathematical Reviews (MathSciNet): MR440724
Zentralblatt MATH: 0291.60029
Digital Object Identifier: doi: 10.1007/BF02392040 - [26] Laurini, F. and Tawn, J. A. (2003). New estimators for the extremal index and other cluster characteristics. Extremes 6 189–211.Mathematical Reviews (MathSciNet): MR2081851
Digital Object Identifier: doi: 10.1023/B:EXTR.0000031179.49454.90 - [27] Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65 291–306.Mathematical Reviews (MathSciNet): MR722133
- [28] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
- [29] Novak, S. Y. (2002). Multilevel clustering of extremes. Stochastic Process. Appl. 97 59–75.Mathematical Reviews (MathSciNet): MR1870720
Zentralblatt MATH: 1060.60049
Digital Object Identifier: doi: 10.1016/S0304-4149(01)00123-5 - [30] Novak, S. Y. (2003). On the accuracy of multivariate compound Poisson approximation. Statist. Probab. Lett. 62 35–43.Mathematical Reviews (MathSciNet): MR1965369
- [31] O’Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Probab. 15 281–289.Mathematical Reviews (MathSciNet): MR877604
Zentralblatt MATH: 0619.60025
Digital Object Identifier: doi: 10.1214/aop/1176992270
Project Euclid: euclid.aop/1176992270 - [32] Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. Astin Bull. 12 22–26.Mathematical Reviews (MathSciNet): MR632572
- [33] Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Probab. 4 529–548.Mathematical Reviews (MathSciNet): MR1272738
Zentralblatt MATH: 0806.60041
Digital Object Identifier: doi: 10.1214/aoap/1177005071
Project Euclid: euclid.aoap/1177005071 - [34] Resnick, S. and Stărică, C. (1998). Tail estimation for dependent data. Ann. Appl. Probab. 8 1156–1183.
- [35] Resnick, S. and Stărică, C. (1999). Smoothing the moment estimator of the extreme value parameter. Extremes 1 263–293.Mathematical Reviews (MathSciNet): MR1814706
Digital Object Identifier: doi: 10.1023/A:1009925716617 - [36] Rootzen, H. (1978). Extremes of moving averages of stable processes. Ann. Probab. 6 847–869.Mathematical Reviews (MathSciNet): MR494450
Zentralblatt MATH: 0394.60025
Digital Object Identifier: doi: 10.1214/aop/1176995432
Project Euclid: euclid.aop/1176995432 - [37] Rootzen, H. (2008). Weak convergence of the tail empirical process for dependent sequences. Stochastic Process. Appl. doi: 10.1016/j.spa.2008.03.003.Mathematical Reviews (MathSciNet): MR2494000
Zentralblatt MATH: 1162.60017
Digital Object Identifier: doi: 10.1016/j.spa.2008.03.003 - [38] Smith, R. L. (1988). A counterexample concerning the extremal index. Adv. in Appl. Probab. 20 681–683.Mathematical Reviews (MathSciNet): MR955511
Zentralblatt MATH: 0652.60044
Digital Object Identifier: doi: 10.2307/1427042
JSTOR: links.jstor.org - [39] Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. Roy. Statist. Soc. Ser. B 56 515–528.
- [40] Shao, Q.-M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.Mathematical Reviews (MathSciNet): MR1415243
Zentralblatt MATH: 0874.60006
Digital Object Identifier: doi: 10.1214/aop/1065725185
Project Euclid: euclid.aop/1041903220 - [41] van der Vaart, A. W. and Wellner, J. (1996). Weak Convergence and Empirical Processes. Springer, New York.
- [42] Vervaat, W. (1972). Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 23 245–253.
- [43] Weissman, I. and Novak, S. Y. (1998). On blocks and runs estimators of the extremal index. J. Statist. Plann. Inference 66 281–288.Mathematical Reviews (MathSciNet): MR1614480
Zentralblatt MATH: 0953.62089
Digital Object Identifier: doi: 10.1016/S0378-3758(97)00095-5 - [44] Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Compound Poisson approximation for the distribution of extremes
Barbour, A. D., Novak, S. Y., and Xia, A., Advances in Applied Probability, 2002 - Model misspecification in peaks over threshold
analysis
Süveges, Mária and Davison, Anthony C., The Annals of Applied Statistics, 2010 - The upcrossings index and the extremal index
Ferreira, H., Journal of Applied Probability, 2006
- Compound Poisson approximation for the distribution of extremes
Barbour, A. D., Novak, S. Y., and Xia, A., Advances in Applied Probability, 2002 - Model misspecification in peaks over threshold
analysis
Süveges, Mária and Davison, Anthony C., The Annals of Applied Statistics, 2010 - The upcrossings index and the extremal index
Ferreira, H., Journal of Applied Probability, 2006 - Asymptotic conditional distribution of exceedance counts
Falk, Michael and Tichy, Diana, Advances in Applied Probability, 2012 - Strong clumping of super-Brownian motion in a stable catalytic medium
Dawson, Donald A., Fleischmann, Klaus, and Mörters, Peter, The Annals of Probability, 2002 - A sliding blocks estimator for the extremal index
Robert, Christian Y., Segers, Johan, and Ferro, Christopher A.T., Electronic Journal of Statistics, 2009 - The Bernoulli sieve revisited
Gnedin, Alexander V., Iksanov, Alexander M., Negadajlov, Pavlo, and Rösler, Uwe, The Annals of Applied Probability, 2009 - Regular variation in the mean and stable limits for Poisson shot noise
Klüppelberg, Claudia, Mikosch, Thomas, and Schärf, Anette, Bernoulli, 2003 - Prediction in a Poisson cluster model
Matsui, Muneya and Mikosch, Thomas, Journal of Applied Probability, 2010 - Extremes of Moving Averages of Stable Processes
Rootzen, Holger, The Annals of Probability, 1978