Fisher–Tippett–Gnedenko theorem

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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)[1], previous versions were stated by Ronald Fisher and Leonard Henry Caleb Tippett (1928)[2] and Fréchet (1927).[3]

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.

Statement[edit]

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.[4] These can be grouped into the generalized extreme value distribution.

Conditions of convergence[edit]

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.

See also[edit]

Notes[edit]

  1. ^ 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
  2. ^ 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
  3. ^ Fréchet, M. (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la societe Polonaise de Mathematique, 6 (1): 93–116
  4. ^ Mood, A.M. (1950). "5. Order Statistics". Introduction to the theory of statistics. New York, NY, US: McGraw-Hill. pp. 251–270.

References[edit]

  • de Haan, Laurens; Ferreira, Ana (2006). Extreme Value Theory: An Introduction. New York: Springer. pp. 6–12. ISBN 0-387-34471-3.