Skip to main content

Generalized Extreme Value Family of Probability Distributions

  • Reference work entry
  • First Online:
International Encyclopedia of Statistical Science
  • 430 Accesses

In real life phenomenon we experience largest observations (maximum extreme) or smallest observation (minimum extreme), such as “How tall should one design an embankment so that the sea reaches this level only once in 100 years?”; “What is the lowest value the Dow Jones Industrial Average can reach in the next three years?”; “How high a drug concentration in the bloodstream can go before causing toxicity?”, among others.

To characterize and understand the behavior of these extremes, we usually use probabilistic extreme value theory. Such theory deals with the stochastic behavior of the minimum and maximum of independent identically distributed random variables. Here, we shall give a brief introduction of the generalized extreme value family of probability distribution that fits the subject observations. This family of probability distribution function (pdf) consists of three famous classical pdfs, namely the Gumbel, the Frechet, and the Weibull.

Extreme value theory has been...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References and Further Reading

  • Abdelhafez MEM, Thomas DR (1990) Approximate prediction limits for the Weibull and extreme value regression models. Egyptian Stat J 34:408–419

    Google Scholar 

  • Achcar JA (1991) A useful reparametrization for the extreme value distribution. Comput Stat Quart 6:113–125

    Google Scholar 

  • Ahmad MI, Sinclair CD, Spurr BD (1988) Assessment of flood frequency models using empirical distribution function statistics. Water Resour Res 24:1323–1328

    Google Scholar 

  • Ahsanullah M (1990) Inference and prediction problems of the Gumbel distribution based on smallest location parameters. Statistician 38:191–195

    Google Scholar 

  • Aitkin M, Clayton D (1980) The fitting of exponential, Weibull, and extreme value distributions to complex censored survival data using GLIM. Appl Stat 29:156–163

    MATH  Google Scholar 

  • Al-Abbasi JN, Fahmi KJ (1991) GEMPAK: a Fortran-77 program for calculating Gumbel’s first, third and mixture upper earthquake magnitude distribution employing maximum likelihood estimation. Comput Geosci 17:271–290

    Google Scholar 

  • Azuz PM (1955) Application of the statistical theory of extreme value to the analysis of maximum pit depth data for aluminum. Corrosion 12:35–46

    Google Scholar 

  • Barnett V (1990) Ranked set sample design for environmental investigations. Environ Ecolog Stat 6:59–74

    Google Scholar 

  • Beran M, Hosking JRM, Arnell N (1986) Comment on “Two-component extreme value distribution for flood analysis” by Fabio Rossi, Mauro Fiorentino. Pasquale Versace Water Resources Res 22:263–266

    Google Scholar 

  • Broussard JP, Booth GG (1998) The behavior of extreme values in Germany’s stock index futures: An application to intradaily margin setting. Eur J Oper Res 104:393–402

    MATH  Google Scholar 

  • Buishand TA (1985) The effect of seasonal variation and serial correlation on the extreme value distribution of rainfall data. J Climate Appl Meteor 25:154–160

    Google Scholar 

  • Buishand TA (1989) Statistics of extremes in climatology. Stat Neerl 43:1–30

    MATH  Google Scholar 

  • Campell JW, Tsokos CP (1973a) The asymptotic distribution of maximum in bivariate samples. J Am Stat Assoc 68:734–739

    Google Scholar 

  • Campbell JW, Tsokos CP (1973b) The asymptotic distribution of maxima in bivariate samples. J Am Stat Assoc 68:734–739

    MATH  MathSciNet  Google Scholar 

  • Changery MJ (1982) Historical extreme winds for the United States-Atlantic and Gulf of Mexico coastlines. U.S. Nuclear Regulatory Commission, NUREG/CR-2639

    Google Scholar 

  • Cheng S, Peng L, Qi Y (1998) Almost sure convergence in extreme value theory. Math Nachr 190:43–50

    MATH  MathSciNet  Google Scholar 

  • Chowbury JU, Stedinger JR, Lu LH (1991) Goodness-of-fit tests for regional generalized extreme value flood distributions. Water Resour Res 27:1765–1776

    Google Scholar 

  • Cohen JP (1986) Large sample theory for fitting an approximating Gumbel model to maxima. Sankhya A 48:372–392

    MATH  Google Scholar 

  • Coles SG, Pan F (1996) The analysis of extreme value pollution levels: A case study. J R Stat 23:333–348

    Google Scholar 

  • Coles SG, Tawn JA (1994) Statistical methods for multivariate extremes: an application to structural design. Appl Stat 43:1–48

    MATH  Google Scholar 

  • Coles SG, Tawn JA (1996) A Bayesian analysis of extreme stock data. Appl Stat 45:463–478

    Google Scholar 

  • Daniels HE (1942) A property of the distribution of extremes. Biometrika 32:194–195

    MathSciNet  Google Scholar 

  • Davidovich MI (1992) On convergence of the Weibull-Gnedenko distribution to the extreme value distribution. Vestnik Akad Nauk Belaruss Ser Mat Fiz, No. 1, Minsk, 103–106

    Google Scholar 

  • De Haan L (1970) On regular variation and its application to the weak convergence of sample extremes. Mathmatical Center Tracts, 32, Mathematisch Centrum. Amsterdam

    Google Scholar 

  • De Hann L, Resnick SI (1998) Sea and wind: Multivariate extreme at work. Extremes 1:7–46

    MathSciNet  Google Scholar 

  • Diebold FX, Schuermann T, Stroughair JD (1999) Pitfalls and opportunities in the use of extreme value theory in risk management. Draft Report

    Google Scholar 

  • Eldredge GG (1957) Analysis of corrosion pitting by extreme value statistics and its application to oil well tubing caliper surveys. Corrosion 13:51–76

    Google Scholar 

  • Embrechts P, Kluppelberg C, Mikosch T (1997) Modeling extremal events for insurance and finance. Spring, Berlin

    Google Scholar 

  • Engelhardt M, Bain LJ (1973) Some complete and censored results for the Weibull or extreme-value distribution. Technometrics 15:541–549

    MATH  MathSciNet  Google Scholar 

  • Engelund S, Rackwitz R (1992) On predictive distribution function for the three asymptotic extreme value distributions. Struct Saf 11:255–258

    Google Scholar 

  • Epstein B (1948) Application to the theory of extreme values in fracture problems. J Am Stat Assoc 43:403–412

    MATH  Google Scholar 

  • Fahmi KJ, Al-Abbasi JN (1991) Application of a mixture distribution of extreme values to earthquake magnitudes in Iraq and conterminous regions. Geophys J R Astron Soc 107:209–217

    Google Scholar 

  • Frechet M (1927) Sur la loi de probabilite de l’ecart maximum. Ann Soc Polon Math Cravovie 6:93–116

    Google Scholar 

  • Frenkel JI, Kontorova TA (1943) A statistical theory of the brittle strength of real crystals. J Phys USSR 7:108–114

    Google Scholar 

  • Fuller WE (1914) Flood flows. Trans Am Soc Civ Eng 77:564

    Google Scholar 

  • Galambos J (1981) Extreme value theory in applied probability. Math Scient 6:13–26

    MATH  MathSciNet  Google Scholar 

  • Galambos J (1987) The asymptotic theory of extreme order statistics, 2nd edn. Krieger, Malabar

    MATH  Google Scholar 

  • Goka T (1993) Application of extreme-value theory to reliability physics of electronic parts and on-orbit single event phenomena. Paper presented at the Conference on Extreme Value Theory and Its Applications, May 2–7, 1993, National Institute of Standards, Gainthersburg

    Google Scholar 

  • Greenwood M (1946) The statistical study of infectious diseases. J R Stat Soc A 109:85–109

    MathSciNet  Google Scholar 

  • Greis NP, Wood EF (1981) Regional flood frequency estimation and network design. Water Resources Res 17:1167–1177

    Google Scholar 

  • Gumbel EJ (1935) Les valeurs extremes des distribution statistuques. Ann l’Inst R Soc London A 221:163–198

    Google Scholar 

  • Gumbel EJ (1941) The return period of flood flows. Ann Math Statist 12:163–190

    MathSciNet  Google Scholar 

  • Gumbel EJ (1945) Floods estimated by probability methods. Engrg News-Record 134:97–101

    Google Scholar 

  • Gumbel EJ (1949a) The Statistical Forecast of Floods. Bulletin No. 15, 1–21, Ohio Water Resources Board

    Google Scholar 

  • Gumbel EJ (1958) Statistics of extremes. Columbia University Press, New York

    MATH  Google Scholar 

  • Gumbel EJ (1962a) Statistical estimation of the endurance limit – an application of extreme-value theory. In: Sarhan AE, Greenberg BG (eds) Contributions to order statistic. Wiley, New York, pp 406–431

    Google Scholar 

  • Gumbel EJ (1962b) Statistical theory of extreme value (main results). In: Sarhan AE, Greenberg BG) Contributions to order statistics, Chapter 6. Wiley, New York

  • Gumbel EJ (1962c) Multivariate extremal distributions. Proceeedings of Session ISI, vol 39:471–475

    MATH  MathSciNet  Google Scholar 

  • Gumbel EJ, Goldstein N (1964) Empirical bivariate extremal distributions. J Am Stat Assoc 59:794–816

    MATH  MathSciNet  Google Scholar 

  • Harris B (1970) Order Statistics and their use in testing and estimation, vol 2. Washington

    Google Scholar 

  • Hassanein KM (1972) Simultaneous estimation of the parameters of the extreme value distribution by sample quantiles. Technometrics 14:63–70

    MATH  Google Scholar 

  • Henery RJ (1984) An extreme-value model for predicting the results of horse reaces. Appl Statist 33:125–133

    Google Scholar 

  • Hisel KW (ed) (1994) Extreme values: floods and droughts. Proceedings of International Conference on Stochastic and Statistical Methods in Hydrology and Environmental Enginerring, vol 1, 1993, Kluwer

    Google Scholar 

  • Hosking JRM (1985) Maximum-likelihood estimation of the parameters of the generalized extreme-value distribution. Appl Stat 34:301–310

    Google Scholar 

  • Hosking JRM, Wallis JR (1988) The effect of intersite dependence on regional flood frequency analysis. Water Resources Res 24:588–600

    Google Scholar 

  • Jain D, Singh VP (1987) Estimating parameters of EV1 distribution for flood frequency analysis. Water Resour Res 23:59–71

    Google Scholar 

  • Jenkinson AF (1969) Statistics of extremes, Technical Note No. 98, World Meteorological Organization, Chapter 5, pp. 183–227

    Google Scholar 

  • Joe H (1994) Multivariate extreme value distributions with applications to environmental data. Canad J Stat Probab Lett 9:75–81

    MathSciNet  Google Scholar 

  • Kimball BF (1955) Practical applications of the theory of extreme values. J Am Stat Assoc 50:517–528

    MATH  Google Scholar 

  • Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 9:245–275

    Google Scholar 

  • Mann NR, Scheduer EM, Fertig KW (1973) A new goodness-of-fit test for the two parameter Weibull or extreme-value distribution with unknown parameters. Comm Stat 2:383–400

    MATH  Google Scholar 

  • Marshall RJ (1983) A spatial-temporal model for storm rainfall. J Hydrol 62:53–62

    Google Scholar 

  • Nisan E (1988) Extreme value distribution in estimation of insurance premiums. ASA Proceedings of Business and Economic Statistics Section, pp 562–566

    Google Scholar 

  • Nordquist JM (1945) Theory of largest values, applied to earthquake magnitude. Trans Am Geophys Union 26:29–31

    Google Scholar 

  • Okubo T, Narita N (1980) On the distribution of extreme winds expected in Japan. National Bureau of Standards Special Publication, 560–561, 12pp

    Google Scholar 

  • Pickands J (1981) Multivariate extreme value distributions. Proceedings of 43rd Session of the ISI. Buenos Aires, vol 49, pp 859–878

    Google Scholar 

  • Pickands J (1986) Statistical inference using extreme order statistics. Ann Stat 3:119–131

    MathSciNet  Google Scholar 

  • von Mises R (1923) Uber die Variationsbreite einer Beobachtungsreihe. Sitzungsber Berlin Math Ges 22:3–8

    Google Scholar 

  • von Mises R (1936) La distribution de las plus grande de n valeurs. Rev Math Union Interbalk 1:141–160. Reproduced in Selected Papers of Richard von Mises, II (1954), Am Math Soc 271–294

    Google Scholar 

  • Weibull W (1939a) A statistical theory of the strength of materials. Ing Vet Akad Handlingar 151

    Google Scholar 

  • Weibull W (1939b) The phenomenon of rupture in solids. Ing Vet Akad Handlingar 153:2

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Tsokos, C.P. (2011). Generalized Extreme Value Family of Probability Distributions. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_427

Download citation

Publish with us

Policies and ethics