In statistics, the Fisher–Tippett–Gnedenko theorem (also the Fisher–Tippett theorem or the extreme value theorem) is a general result in extreme value theory regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization can only converge in distribution to one of 3 possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem (or convergence to types theorem) is given to Gnedenko (1948), previous versions were stated by Ronald Fisher and Leonard Henry Caleb Tippett (1928) and Fréchet (1927).
The role of the extremal types theorem for maxima is similar to that of central limit theorem for averages, except that the central limit theorem applies to the average of a sample from any distribution with finite variance, while the Fisher–Tippet–Gnedenko theorem only states that if the distribution of a normalized maximum converges, then the limit has to be one of a particular class of distributions. It does not state that the distribution of the normalized maximum does converge.
Let be a sequence of independent and identically-distributed random variables, and . If a sequence of pairs of real numbers exists such that each and , where is a non-degenerate distribution function, then the limit distribution belongs to either the Gumbel, the Fréchet or the Weibull family. These can be grouped into the generalized extreme value distribution.
Conditions of convergence
If G is the distribution function of X, then Mn can be rescaled to converge in distribution to
- a Fréchet if and only if G (x) < 1 for all real x and . In this case, possible sequences are
- bn = 0 and
- a Weibull if and only if and . In this case possible sequences are
- bn = ω and
Convergence conditions for the Gumbel distribution are more involved.
- Extreme value theory
- Gumbel distribution
- Generalized extreme value distribution
- Pickands–Balkema–de Haan theorem
- Generalized Pareto distribution
- Exponentiated generalized Pareto distribution
- Gnedenko, B.V. (1943), "Sur la distribution limite du terme maximum d'une serie aleatoire", Annals of Mathematics, 44 (3): 423–453, doi:10.2307/1968974, JSTOR 1968974
- Fisher, R.A.; Tippett, L.H.C. (1928), "Limiting forms of the frequency distribution of the largest and smallest member of a sample", Proc. Camb. Phil. Soc., 24 (2): 180–190, Bibcode:1928PCPS...24..180F, doi:10.1017/s0305004100015681
- Fréchet, M. (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la societe Polonaise de Mathematique, 6 (1): 93–116
- Mood, A.M. (1950). "5. Order Statistics". Introduction to the theory of statistics. New York, NY, US: McGraw-Hill. pp. 251–270.
- de Haan, Laurens; Ferreira, Ana (2006). Extreme Value Theory: An Introduction. New York: Springer. pp. 6–12. ISBN 0-387-34471-3.
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