Regression to the tail
Definition
In statistics, regression to the tail is the phenomenon that arises if a distribution has non-vanishing probability density towards infinity.[1] The incidence of new large values and how much they exceed earlier ones determine how fat-tailed a distribution will be, and above a certain frequency and size of values, the mean increases with more events measured, with the mean eventually approaching infinity instead of converging. In this case, regression to the mean means regression to infinity, i.e., a non-existent mean.
Regression to the tail contrasts with the well-known phenomenon "regression to the mean."[2] Where regression to the mean applies for thin-tailed distributions like the Gaussian, regression to the tail applies for fat-tailed distributions like the Pareto distribution and power-law distributions.
To avoid making incorrect inferences, regression to the tail must be considered when designing scientific studies, including measurement, analysis, and interpretation of data. Specifically, it must be decided whether data follow regression to the mean or regression to the tail, and it must be ensured that researchers do not make the error of assuming one when, in fact, the other applies.
The law of regression to the tail
A phenomenon is said to follow the "law of regression to the tail" if events in the relevant distribution appear in the tail with a size and frequency that does not allow the mean or variance to converge. The law depicts a situation with many extreme events, and regardless of how extreme the most extreme event is, there will eventually be an event that is even more extreme than this. It is only a matter of time until it happens.
Examples
Earthquakes prototypically follow the law of regression to the tail,[3] as do pandemics,[4] floods,[5] droughts, wildfires, landslides, avalanches, tsunamis, wars,[6] crime, terrorist attacks, blackouts, financial markets, debt, bankruptcies,[7] and cybercrime.[8]
Coinage
The terms "regression to the tail" and the "law of regression to the tail" were first coined in a comment on the covid-19 pandemic and the climate crisis authored by Professor Bent Flyvbjerg, University of Oxford, published in Salon July 2020[9] and later elaborated and extended in an academic paper published in Environmental Science and Policy October 2020.[10]
References
- ↑ Flyvbjerg, Bent (2020). "The Law of Regression to the Tail: How to Survive Covid-19, the Climate Crisis, and Other Disasters". Environmental Science and Policy. 114: 614–618. doi:10.1016/j.envsci.2020.08.013. PMC 7533687 Check
|pmc=value (help). PMID 33041651 Check|pmid=value (help). - ↑ Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge: Cambridge University Press. ISBN 0-521-81099-X. Search this book on
- ↑ Clauset, A., Shalizi, C. R., and Newman, M. E. (2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51 (4): 661–703. arXiv:0706.1062. Bibcode:2009SIAMR..51..661C. doi:10.1137/070710111. Unknown parameter
|s2cid=ignored (help)CS1 maint: Multiple names: authors list (link) - ↑ Cirillo, Pasquale and Nassim Nicholas Taleb (2020). "Tail Risk of Contagious Diseases". Nature Physics. 16 (6): 606–613. arXiv:2004.08658. Bibcode:2020NatPh..16..606C. doi:10.1038/s41567-020-0921-x. Unknown parameter
|s2cid=ignored (help) - ↑ Malamud, B. D. and Turcotte, D. L. (2006). "The Applicability of Power-Law Frequency Statistics to Floods". Journal of Hydrology. 322 (1–4): 168–180. Bibcode:2006JHyd..322..168M. doi:10.1016/j.jhydrol.2005.02.032.CS1 maint: Multiple names: authors list (link)
- ↑ Newman, M. E. (2005). "Power Laws, Pareto Distributions and Zipf's Law". Contemporary Physics. 46 (5): 323–351. arXiv:cond-mat/0412004. Bibcode:2005ConPh..46..323N. doi:10.1080/00107510500052444. Unknown parameter
|s2cid=ignored (help) - ↑ Hong, B. H., Lee, K. E. and Lee, J. W. (2007). "Power Law in Firms Bankruptcy". Physics Letters A. 361 (1–2): 6–8. arXiv:physics/0701302. Bibcode:2007PhLA..361....6H. doi:10.1016/j.physleta.2006.09.034. Unknown parameter
|s2cid=ignored (help)CS1 maint: Multiple names: authors list (link) - ↑ Maillart, T. and Sornette, D. (2010). "Heavy-Tailed Distribution of Cyber-Risks". The European Physical Journal B. 75 (3): 357–364. Bibcode:2010EPJB...75..357M. doi:10.1140/epjb/e2010-00120-8. hdl:20.500.11850/18658. Unknown parameter
|s2cid=ignored (help)CS1 maint: Multiple names: authors list (link) - ↑ Flyvbjerg, Bent (2020). "Regression to the Tail: How to Mitigate COVID-19, the Climate Crisis, and Other Catastrophes". Salon. July 5.
- ↑ Flyvbjerg, Bent (2020). "The Law of Regression to the Tail: How to Survive Covid-19, the Climate Crisis, and Other Disasters". Environmental Science and Policy. October: 614–618. doi:10.1016/j.envsci.2020.08.013. PMC 7533687 Check
|pmc=value (help). PMID 33041651 Check|pmid=value (help). SSRN 3600070 Check|ssrn=value (help).
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