Model misspecification in peaks over threshold analysis. (English) Zbl 1189.62086

Süveges, Mária; Davison, Anthony C.

Ann. Appl. Stat. 4, No. 1, 203-221 (2010).

Summary: Classical peaks over threshold analysis is widely used for statistical modeling of sample extremes, and can be supplemented by a model for the sizes of clusters of exceedances. Under mild conditions a compound Poisson process model allows the estimation of the marginal distribution of threshold exceedances and of the mean cluster size, but requires the choice of a threshold and of a run parameter, \(K\), that determines how exceedances are declustered. We extend a class of estimators of the reciprocal mean cluster size, known as the extremal index, establish consistency and asymptotic normality, and use the compound Poisson process to derive misspecification tests of model validity and of the choice of the run parameter and threshold. Simulated examples and real data on temperatures and rainfall illustrate the ideas, both for estimating the extremal index in nonstandard situations and for assessing the validity of extremal models.

MSC:

62G32	Statistics of extreme values; tail inference
62M99	Inference from stochastic processes
65C60	Computational problems in statistics (MSC2010)
62P12	Applications of statistics to environmental and related topics
62E20	Asymptotic distribution theory in statistics

Keywords:

cluster; extremal index; extreme value theory; likelihood; model misspecification; Neuchâtel temperature data; Venezuelan rainfall data

Software:

ismev

References:

[1]	Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes . Wiley, Chichester. · Zbl 1070.62036 · doi:10.1002/0470012382
[2]	Beirlant, J., Vynckier, P. and Teugels, J. L. (1996). Excess functions and estimation of the extreme-value index. Bernoulli 2 293-318. · Zbl 0870.62019 · doi:10.2307/3318416
[3]	Benjamini, Y. and Hochberg, Y. (1995). Controlling the False Discovery Rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289-300. · Zbl 0809.62014
[4]	Benjamini, Y. and Yekutieli, D. (2001). The control of the False Discovery Rate in multiple testing under dependency. Ann. Statist. 29 1165-1188. · Zbl 1041.62061 · doi:10.1214/aos/1013699998
[5]	Brown, S. J., Caesar, J. and Ferro, C. A. T. (2008). Global changes in extreme daily temperature since 1950. J. Geophys. Res. 113 D05115. doi: .
[6]	Coles, S. G. (2001). An Introduction to Statistical Modeling of Extreme Values . Springer, London. · Zbl 0980.62043
[7]	Coles, S. G. and Pericchi, L. (2003). Anticipating catastrophes through extreme-value modelling. Appl. Statist. 52 405-416. · Zbl 1111.62366 · doi:10.1111/1467-9876.00413
[8]	Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds (with discussion). J. Roy. Statist. Soc. Ser. B 52 393-442. · Zbl 0706.62039
[9]	Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modeling Extremal Events for Insurance and Finance . Springer, Berlin. · Zbl 0873.62116
[10]	Falk, M., Hüsler, J. and Reiss, D. (2004). Laws of Small Numbers: Extremes and Rare Events . Birkhäuser, Basel. · Zbl 1083.60003
[11]	Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . Chapman & Hall, London. · Zbl 0873.62037
[12]	Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 65 545-556. · Zbl 1065.62091 · doi:10.1111/1467-9868.00401
[13]	Hsing, T. (1987). On the characterization of certain point processes. Stochastic Process. Appl. 26 297-316. · Zbl 0645.60057 · doi:10.1016/0304-4149(87)90183-9
[14]	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
[15]	Kharin, V. V. and Zwiers, F. W. (2005). Estimating extremes in transient climate change simulations. Journal of Climate 18 1156-1173.
[16]	Larsen, M. C., Wieczorek, G. F., Eaton, L. S., Morgan, B. A. and Torres-Sierra, H. (2001). Venezuelan debris flow and flash flood disaster of 1999 studied. EOS, Transactions of the American Geophysical Union 47 572.
[17]	Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes . Springer, New York. · Zbl 0518.60021
[18]	Nogaj, M., Yiou, P., Parey, S., Malek, F. and Naveau, P. (2006). Amplitude and frequency of temperature extremes over the North Atlantic region. Geophys. Res. Lett. 33 L10801. doi: .
[19]	Pickands, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119-131. · Zbl 0312.62038 · doi:10.1214/aos/1176343003
[20]	Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes . Springer, New York. · Zbl 0633.60001
[21]	Smith, R. L. (1985). Maximum likelihood estimation in a class of non-regular cases. Biometrika 72 67-90. · Zbl 0583.62026 · doi:10.1093/biomet/72.1.67
[22]	Smith, R. L. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone. Statist. Sci. 4 367-377. · Zbl 0955.62646 · doi:10.1214/ss/1177012400
[23]	Smith, R. L. (1992). The extremal index for a Markov chain. J. Appl. Probab. 29 37-45. · Zbl 0759.60059 · doi:10.2307/3214789
[24]	Sousa, B. and Michailidis, G. (2004). A diagnostic plot for estimating the tail index of a distribution. J. Comput. Graph. Statist. 13 974-995. · doi:10.1198/106186004X12335
[25]	Süveges, M. (2007). Likelihood estimation of the extremal index. Extremes 10 41-55. · Zbl 1150.62368 · doi:10.1007/s10687-007-0034-2
[26]	White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50 1-25. · Zbl 0478.62088 · doi:10.2307/1912526
[27]	White, H. (1994). Estimation, Inference and Specification Analysis . Cambridge Univ. Press, Cambridge. · Zbl 0860.62100