8 Further Reading

This section contains references on extremes which may not be particularly related to the individual topics in the notes, and/or fall into multiple categories. While they are mostly out of the scope of this module, they are good starting points for pursuing further research in the field.

8.1 General & books

Hill (1975) and Pickands (1975), which were published in the same journal and year back in 1975, are two of the papers that paved the way for classical theory of extreme value theory. Geffroy (1958/59) is another old publication on extremes.

Introductory books to the topic include J. Beirlant et al. (2004) and S. Coles (2001), the latter of which is accompanied by the R package ismev that you have used in the labs.

This paper by Jan Beirlant, Caeiro, and Gomes (2012) lists the open problems (at least they were a few years ago) in univariate extremes. This review by Chavez-Demoulin and Davison (2012) covers various topics introduced in this module, albeit titled “Modelling time series extremes”.

8.2 Environmental applications

There are a lot of applications to environmental data, and we list a few areas here:

• Climate change and/or downscaling: Burke, Perry, and Brown (2010), Mannshardt-Shamseldin et al. (2010), Kallache et al. (2011), Winter, Tawn, and Brown (2016), Shen, Mickley, and Gilleland (2016)
• Ocean extremes: Jonathan and Ewans (2007), Jonathan and Ewans (2013), Mackay and Jonathan (2020)
• Flooding: Keef, Tawn, and Svensson (2009), Keef, Tawn, and Lamb (2013)
• This blog post discusses how we quantify risk from environmental extreme events in a changing world, and contains some additional references.

Bibliography

Beirlant, Jan, Frederico Caeiro, and M. Ivette Gomes. 2012. “An Overview and Open Research Topics in Statistics of Univariate Extremes.” REVSTAT – Statistical Journal 10 (1): 1–31. https://www.ine.pt/revstat/autores/pdf/rs120101.pdf.

Beirlant, J., Y. Goegebeur, J. Segers, and J. Teugels. 2004. Statistics of Extremes: Theory and Applications. Chichester: Wiley.

Burke, Eleanor J., Richard H. J. Perry, and Simon J. Brown. 2010. “An Extreme Value Analysis of UK Drought and Projections of Change in the Future.” Journal of Hydrology 388 (1): 131–43. https://doi.org/https://doi.org/10.1016/j.jhydrol.2010.04.035.

———. 2012. “Modelling Time Series Extremes.” REVSTAT – Statistical Journal 10 (1): 109–33. https://infoscience.epfl.ch/record/180506/files/rs120105.pdf.

Coles, Stuart. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer.

Geffroy, J. 1958/59. “Contributions à La Théorie Des Valeurs Extrème.” Publ. Inst. Statist. Univ. Paris 7–8 (1958/59): 37–185.

Hill, Bruce M. 1975. “A Simple General Approach to Inference about the Tail of a Distribution.” The Annals of Statistics 3 (5): 1163–74.

Jonathan, Philip, and Kevin Ewans. 2007. “The Effect of Directionality on Extreme Wave Design Criteria.” Ocean Engineering 34 (14): 1977–94. https://doi.org/https://doi.org/10.1016/j.oceaneng.2007.03.003.

———. 2013. “Statistical Modelling of Extreme Ocean Environments for Marine Design: A Review.” Ocean Engineering 62: 91–109. https://doi.org/https://doi.org/10.1016/j.oceaneng.2013.01.004.

Kallache, M., M. Vrac, P. Naveau, and P.-A. Michelangeli. 2011. “Nonstationary Probabilistic Downscaling of Extreme Precipitation.” Journal of Geophysical Research: Atmospheres 116 (D5). https://doi.org/https://doi.org/10.1029/2010JD014892.

Keef, Caroline, Jonathan A. Tawn, and Rob Lamb. 2013. “Estimating the Probability of Widespread Flood Events.” Environmetrics 24 (1): 13–21. https://doi.org/https://doi.org/10.1002/env.2190.

Keef, Caroline, Jonathan Tawn, and Cecilia Svensson. 2009. “Spatial Risk Assessment for Extreme River Flows.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 58 (5): 601–18. https://doi.org/https://doi.org/10.1111/j.1467-9876.2009.00672.x.

Mackay, Ed, and Philip Jonathan. 2020. “Assessment of Return Value Estimates from Stationary and Non-Stationary Extreme Value Models.” Ocean Engineering 207: 107406. https://doi.org/https://doi.org/10.1016/j.oceaneng.2020.107406.

Mannshardt-Shamseldin, Elizabeth C., Richard L. Smith, Stephan R. Sain, Linda O. Mearns, and Daniel Cooley. 2010. “Downscaling Extremes: A Comparison of Extreme Value Distributions in Point-Source and Gridded Precipitation Data.” The Annals of Applied Statistics 4 (1): 484–502. http://www.jstor.org/stable/27801596.

Pickands, J. 1975. “Statistical Inference Using Extreme Order Statistics.” The Annals of Statistics 3 (1): 119–31.

Shen, L., L. J. Mickley, and E. Gilleland. 2016. “Impact of Increasing Heat Waves on U.S. Ozone Episodes in the 2050s: Results from a Multimodel Analysis Using Extreme Value Theory.” Geophysical Research Letters 43 (8): 4017–25. https://doi.org/https://doi.org/10.1002/2016GL068432.

Winter, Hugo C., Jonathan A. Tawn, and Simon J. Brown. 2016. “Modelling the Effect of the El Niño-Southern Oscillation on Extreme Spatial Temperature Events over Australia.” Annals of Applied Statistics 10 (4): 2075–2101. https://doi.org/10.1214/16-AOAS965.