Extreme value theory

On the decay of awfulness with oftenness

In a satisfying way, it turns out that there are only so many shapes that probability densities can assume as they head off toward infinity. Extreme value theory makes this precise, and gives us some tools to work with them.

See also densities and intensities, survival analysis.

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Balkema, A. A., and L. de Haan. 1974. “Residual Life Time at Great Age.” The Annals of Probability 2 (5): 792–804. https://doi.org/10.1214/aop/1176996548.

Bhatti, Sajjad Haider, Shahzad Hussain, Tanvir Ahmad, Muhammad Aslam, Muhammad Aftab, and Muhammad Ali Raza. 2018. “Efficient Estimation of Pareto Model: Some Modified Percentile Estimators.” PLOS ONE 13 (5): e0196456. https://doi.org/10.1371/journal.pone.0196456.

Castillo, Enrique, and Ali S. Hadi. 1997. “Fitting the Generalized Pareto Distribution to Data.” Journal of the American Statistical Association 92 (440): 1609–20. https://doi.org/10.1080/01621459.1997.10473683.

Charpentier, Arthur, and Emmanuel Flachaire. 2019. “Pareto Models for Risk Management,” December. http://arxiv.org/abs/1912.11736.

Dargahi-Noubary, G. R. 1989. “On Tail Estimation: An Improved Method.” Mathematical Geology 21 (8): 829–42. https://doi.org/10.1007/BF00894450.

Embrechts, Paul, S Kluppelberg, and Thomas Mikosch. 1997. Extremal Events in Finance and Insurance. Springer Berlin Heidelberg.

Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch. 1997. “Risk Theory.” In Modelling Extremal Events, 21–57. Applications of Mathematics 33. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-642-33483-2_2.

Fisher, R. A., and L. H. C. Tippett. 1928. “Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample.” Mathematical Proceedings of the Cambridge Philosophical Society 24 (2): 180–90. https://doi.org/10.1017/S0305004100015681.

Ghitany, Mohamed, Emilio Gómez-Déniz, and Saralees Nadarajah. 2018. “A New Generalization of the Pareto Distribution and Its Application to Insurance Data.” Journal of Risk and Financial Management 11 (1): 10. https://doi.org/10.3390/jrfm11010010.

Makarov, Mikhail. 2006. “Extreme Value Theory and High Quantile Convergence.” The Journal of Operational Risk 1 (2): 51–57. https://doi.org/10.21314/JOP.2006.009.

Markovitch, Natalia M, and Udo R Krieger. 2002. “The Estimation of Heavy-Tailed Probability Density Functions, Their Mixtures and Quantiles.” Computer Networks 40 (3): 459–74. https://doi.org/10.1016/S1389-1286(02)00306-7.

McNeil, Alexander J. 1997. “Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory.” ASTIN Bulletin: The Journal of the IAA 27 (1): 117–37. https://doi.org/10.2143/AST.27.1.563210.

McNeil, Alexander J, Rüdiger Frey, and Paul Embrechts. 2005. Quantitative Risk Management : Concepts, Techniques and Tools. Princeton: Princeton Univ. Press.

Mueller, Ulrich K. 2018. “Refining the Central Limit Theorem Approximation via Extreme Value Theory,” February. http://arxiv.org/abs/1802.00762.

Naveau, Philippe, Alexis Hannart, and Aurélien Ribes. 2020. “Statistical Methods for Extreme Event Attribution in Climate Science.” Annual Review of Statistics and Its Application 7 (1): 89–110. https://doi.org/10.1146/annurev-statistics-031219-041314.

Pickands III, James. 1975. “Statistical Inference Using Extreme Order Statistics.” The Annals of Statistics 3 (1): 119–31. https://doi.org/10.1214/aos/1176343003.

Vajda, S. 1951. “Analytical Studies in Stop-Loss Reinsurance.” Scandinavian Actuarial Journal 1951 (1-2): 158–75. https://doi.org/10.1080/03461238.1951.10432137.

Wong, T. S. T., and W. K. Li. 2006. “A Note on the Estimation of Extreme Value Distributions Using Maximum Product of Spacings.” In Institute of Mathematical Statistics Lecture Notes - Monograph Series, 272–83. Beachwood, Ohio, USA: Institute of Mathematical Statistics. http://projecteuclid.org/euclid.lnms/1196285981.

Zhao, Xu, Zhongxian Zhang, Weihu Cheng, and Pengyue Zhang. 2019. “A New Parameter Estimator for the Generalized Pareto Distribution Under the Peaks over Threshold Framework.” Mathematics 7 (5): 406. https://doi.org/10.3390/math7050406.