Fisher–Tippett–Gnedenko theorem

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This article is about the extreme value theorem in statistics. For the result in calculus, see extreme value theorem.

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 Fisher and Tippett in 1928 and Fréchet in 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.

Statement

Let (X_{1},X_{2}\ldots ,X_{n}) be a sequence of independent and identically-distributed random variables, let M_{n}=\max\{X_{1},\ldots ,X_{n}\}. If a sequence of pairs of real numbers (a_{n},b_{n}) exists such that each a_{n}>0 and \lim _{{n\to \infty }}P\left({\frac {M_{n}-b_{n}}{a_{n}}}\leq x\right)=F(x), where F is a non degenerate distribution function, then the limit distribution F belongs to either the Gumbel, the Fréchet or the Weibull family. These distributions 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 law to

  • a Fréchet if and only if G (x) < 1 for all real x and {\frac {1-G(tx)}{1-G(t)}}{\xrightarrow[ {t\to +\infty }]{}}x^{{-\theta }},\quad x>0. In this case, possible sequences are
bn = 0 and a_{n}=G^{{-1}}\left(1-{\frac {1}{n}}\right).
  • a Weibull if and only if \omega =\sup\{G<1\}<+\infty and {\frac {1-G(\omega +tx)}{1-G(\omega -t)}}{\xrightarrow[ {t\to 0^{+}}]{}}(-x)^{\theta },\quad x<0. In this case possible sequences are
bn = ω and a_{n}=\omega -G^{{-1}}\left(1-{\frac {1}{n}}\right).

See also


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