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Robert, Christian Y.

Inference for the limiting cluster size distribution of extreme values. (English) Zbl 1158.62061

Ann. Stat. 37, No. 1, 271-310 (2009).
Summary: Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily a 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.
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.

Cited in 23 Documents

MSC:

62M99 Inference from stochastic processes
60G70 Extreme value theory; extremal stochastic processes
62G32 Statistics of extreme values; tail inference
62E20 Asymptotic distribution theory in statistics
62M09 Non-Markovian processes: estimation
62G20 Asymptotic properties of nonparametric inference

Keywords:

exceedance point processes; extremal index; strictly stationary sequences
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Full Text: DOI arXiv Euclid

References:

[1] Ancona-Navarrete, M. and Tawn, J. A. (2000). A comparison of methods for estimating the extremal index. Extremes 3 5-38. · Zbl 0965.62044 · doi:10.1023/A:1009993419559
[2] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[3] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003
[4] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley, New York. · Zbl 0822.60002
[5] Buchmann, B. and Grübel, R. (2003). Decompounding: An estimation problem for Poisson random sums. Ann. Statist. 31 1054-1074. · Zbl 1105.62309 · doi:10.1214/aos/1059655905
[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. · Zbl 1078.62020 · doi:10.1007/BF02506487
[7] Deo, C. M. (1973). A note on empirical processes of strong-mixing sequences. Ann. Probab. 5 870-875. · Zbl 0281.60034 · doi:10.1214/aop/1176996855
[8] Doukhan, P. (1994). Mixing. Properties and Examples . Springer, New York. · Zbl 0801.60027
[9] Drees, H. (1998). On smooth statistical tail functionals. Scand. J. Statist. 25 187-210. · Zbl 0923.62032 · doi:10.1111/1467-9469.00097
[10] Drees, H. (2000). Weighted approximations of tail processes for \beta -mixing random variables. Ann. Appl. Probab. 10 1274-1301. · Zbl 1073.60520 · doi:10.1214/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. · Zbl 1021.62038
[12] Drees, H. (2003). Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9 617-657. · Zbl 1040.62077 · doi:10.3150/bj/1066223272
[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events . Springer, Berlin. · Zbl 0873.62116
[14] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987-1008. JSTOR: · Zbl 0491.62099 · doi:10.2307/1912773
[15] Ferro, C. A. T. (2003). Statistical methods for clusters of extreme values. Ph.D. thesis, Lancaster Univ. · Zbl 1065.62091
[16] Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 65 545-556. JSTOR: · Zbl 1065.62091 · doi:10.1111/1467-9868.00401
[17] Goldie, C. M. (1989). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166. · Zbl 0724.60076 · doi:10.1214/aoap/1177005985
[18] de Haan, L. and Resnick, S. (1993). Estimating the limiting distribution of multivariate extremes. Commun. Statist. Stochatic Models 9 275-309. · Zbl 0777.62036 · 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. · Zbl 0679.60029 · 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. · Zbl 0722.62021 · 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. · Zbl 0797.62018 · doi:10.1214/aos/1176349409
[23] Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Probab. Theory Related Fields 95 331-356. · Zbl 0791.60024 · 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. · Zbl 0619.60054 · doi:10.1007/BF00718038
[25] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248. · Zbl 0291.60029 · 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. · Zbl 1053.62065 · 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. · Zbl 0506.60030 · doi:10.1007/BF00532484
[28] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes . Springer, New York. · Zbl 0518.60021
[29] Novak, S. Y. (2002). Multilevel clustering of extremes. Stochastic Process. Appl. 97 59-75. · Zbl 1060.60049 · 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. · Zbl 1101.62310 · doi:10.1016/S0167-7152(02)00422-4
[31] O’Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Probab. 15 281-289. · Zbl 0619.60025 · doi:10.1214/aop/1176992270
[32] Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. Astin Bull. 12 22-26.
[33] Perfekt, R. (1994). Extremal behaviour of stationary Markov chains with applications. Ann. Appl. Probab. 4 529-548. · Zbl 0806.60041 · doi:10.1214/aoap/1177005071
[34] Resnick, S. and Stărică, C. (1998). Tail estimation for dependent data. Ann. Appl. Probab. 8 1156-1183. · Zbl 0942.60037 · doi:10.1214/aoap/1028903376
[35] Resnick, S. and Stărică, C. (1999). Smoothing the moment estimator of the extreme value parameter. Extremes 1 263-293. · Zbl 0921.62028 · doi:10.1023/A:1009925716617
[36] Rootzen, H. (1978). Extremes of moving averages of stable processes. Ann. Probab. 6 847-869. · Zbl 0394.60025 · doi:10.1214/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. · Zbl 1162.60017 · 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. JSTOR: · Zbl 0652.60044 · doi:10.2307/1427042
[39] Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. Roy. Statist. Soc. Ser. B 56 515-528. JSTOR: · Zbl 0796.62084
[40] Shao, Q.-M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098-2127. · Zbl 0874.60006 · doi:10.1214/aop/1041903220
[41] van der Vaart, A. W. and Wellner, J. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[42] Vervaat, W. (1972). Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 23 245-253. · Zbl 0238.60018 · doi:10.1007/BF00532510
[43] Weissman, I. and Novak, S. Y. (1998). On blocks and runs estimators of the extremal index. J. Statist. Plann. Inference 66 281-288. · Zbl 0953.62089 · doi:10.1016/S0378-3758(97)00095-5
[44] Whitt, W. (2002). Stochastic-Process Limits . Springer, New York. · Zbl 0993.60001 · doi:10.1007/b97479
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