Abstract
There are some suggestions that extreme weather events are becoming more frequent due to global warming. From a statistical point of view, this raises the question of trend detection in the extremes of a series of observations. We build upon the heteroscedastic extremes framework by Einmahl et al. (J. R. Stat. Soc. Ser. B. Stat. Methodol. 78(1), 31–51, 2016) where the observations are assumed independent but not identically distributed and the variation in their tail distributions is modeled by the so-called skedasis function. While the original paper focuses on non parametric estimation of the skedasis function, we consider here parametric models and prove the consistency and asymptotic normality of the parameter estimators. A parametric test for trend detection in the case where the skedasis function is monotonic is introduced. A short simulation study shows that the parametric test can be more powerful than the non parametric Kolmogorov-Smirnov type test, even for misspecified models. The methodology is finally illustrated on a dataset of daily maximal temperatures in Fort Collins, Colorado, during the 20th century.
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References
Barndorff-Nielsen, O.: Information and Exponential Families in Statistical Theory. Wiley, Chichester (1978). Wiley Series in Probability and Mathematical Statistics
Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer, London (2001)
Daouia, A., Gardes, L., Girard, S.: On kernel smoothing for extremal quantile regression. Bernoulli 19(5B), 2557–2589 (2013)
Davison, A., Smith, R.: Models for exceedances over high thresholds. J. R. Stat. Soc. Ser. B. Stat. Methodol. 52(3), 393–442 (1990). With discussion and a reply by the authors
de Haan, L., Ferreira, A.: Extreme Value Theory. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006). An introduction
de Haan, L., Klein Tank, A., Neves, C.: On tail trend detection: Modeling relative risk. Extremes 18(2), 141–178 (2015)
Drees, H., Rootzén, H.: Limit theorems for empirical processes of cluster functionals. Ann Statist. 38(4), 2145–2186 (2010)
Einmahl, J., de Haan, L., Zhou, C.: Statistics of heteroscedastic extremes. J. R. Stat. Soc. Ser. B. Stat Methodol. 78(1), 31–51 (2016)
Embrechts, P., Goldie, C. M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49(3), 335–347 (1979)
Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events. Birkhäuser/Springer, Basel (2011). Basel extended edition
Gardes, L.: A general estimator for the extreme value index: Applications to conditional and heteroscedastic extremes. Extremes 18(3), 479–510 (2015)
Gardes, L., Girard, S.: Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels. Extremes 13(2), 177–204 (2010)
Groisman, P., Knight, R., Easterling, D., Karl, T., Hegerl, G., Razuvaev, V.: Trends in intense precipitation in the climate record. J. Climate 18 (9), 1326–1350 (2005)
Hsing, T.: On tail index estimation using dependent data. Ann. Statist. 19(3), 1547–1569 (1991)
Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publ. Inst Math. (Beograd) (N.S.) 80(94), 121–140 (2006)
Kallenberg, O.: Random Measures, Theory and Applications, Volume 77 of Probability Theory and Stochastic Modelling. Springer, Cham (2017)
Klein Tank, A., Können, G.: Trends in indices of daily temperature and precipitation extremes in Europe, 1946-99. J. Climate 16(22), 3665–3680 (2003)
Leadbetter, M., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. Springer, New York (1983)
McElroy, T.: On the measurement and treatment of extremes in time series. Extremes 19(3), 467–490 (2016)
Resnick, S.: Extreme Values, Regular Variation, and Point Processes, Volume 4 of Applied Probability. A Series of the Applied Probability Trust. Springer, New York (1987)
Seshadri, V., Csorgo, M., Stephens, M.A.: Tests for the exponential distribution using Kolmogorov-type statistics. J. R. Stat. Soc. Ser. B. Stat Methodol. 31(3), 499–509 (1969)
van der Vaart, A.: Asymptotic Statistics, Volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998)
Zolina, O., Simmer, C., Belyaev, K., Kapala, A., Gulev, S.: Improving estimates of heavy and extreme precipitation using daily records from European rain gauges. J. Hydrometeorol. 10(3), 701–716 (2009)
Acknowledgments
The research by A.Mefleh is financed by a joint funding program from the Lebanese National Council for Scientific Research CNRS-L and the Lebanese University LU. The research of Clé ment Dombry is partially supported by the Bourgogne Franche-Comté region (grant OPE-2017-0068).
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Mefleh, A., Biard, R., Dombry, C. et al. Trend detection for heteroscedastic extremes. Extremes 23, 85–115 (2020). https://doi.org/10.1007/s10687-019-00363-1
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DOI: https://doi.org/10.1007/s10687-019-00363-1