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 and its convergence details are given to Fréchet (1927),[1] Ronald Fisher and Leonard Henry Caleb Tippett (1928),[2] Mises (1936)[3][4] and Gnedenko (1943).[5]

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 with cumulative distribution function . Suppose that there exist two sequences of real numbers and such that the following limits converge to a non-degenerate distribution function:

,

or equivalently:

.

In such circumstances, the limit distribution belongs to either the Gumbel, the Fréchet or the Weibull family.[6]

In other words, if the limit above converges we will have assume the form:[7]

or else

for some parameter This is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index . The GEV distribution groups the Gumbel, Fréchet and Weibull distributions into a single one. Note that the second formula (the Gumbel distribution) is the limit of the first as goes to zero.

Conditions of convergence[edit]

The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution above. The study of conditions for convergence of to particular cases of the generalized extreme value distribution began with Mises, R. (1936)[3][5][4] and was further developed by Gnedenko, B. V. (1943).[5]

Let be the distribution function of , and an i.i.d. sample thereof. Also let be the populational maximum, i.e. . The limiting distribution of the normalized sample maximum, given by above, will then be:[7]

  • A Fréchet distribution () if and only if and for all .
This corresponds to what we may call a heavy tail. In this case, possible sequences that will satisfy the theorem conditions are and .
  • A Gumbel distribution (), with finite or infinite, if and only if for all with .
Possible sequences here are and .
  • A Weibull distribution () if and only if is finite and for all .
Possible sequences here are and .

Examples[edit]

Fréchet distribution[edit]

If we take the Cauchy distribution

the cumulative distribution function is:

is asymptotic to or

and we have

Thus we have

and letting (and skipping some explanation)

for any The expected maximum value therefore goes up linearly with n.

Gumbel distribution[edit]

Let us take the normal distribution with cumulative distribution function

We have

and

Thus we have

If we define as the value that satisfies

then around

As n increases, this becomes a good approximation for a wider and wider range of so letting we find that

We can see that and then

so the maximum is expected to climb ever more slowly toward infinity.

Weibull distribution[edit]

We may take the simplest example, a uniform distribution between 0 and 1, with cumulative distribution function

from 0 to 1.

Approaching 1 we have

Then

Letting we have

The expected maximum approaches 1 inversely proportionally to n.

See also[edit]

Notes[edit]

  1. ^ Fréchet, M. (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la Société Polonaise de Mathématique, 6 (1): 93–116
  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. ^ a b Mises, R. von (1936). "La distribution de la plus grande de n valeurs". Rev. Math. Union Interbalcanique 1: 141–160.
  4. ^ a b Falk, Michael; Marohn, Frank (1993). "Von Mises conditions revisited". The Annals of Probability: 1310–1328.
  5. ^ a b c 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
  6. ^ Mood, A.M. (1950). "5. Order Statistics". Introduction to the theory of statistics. New York, NY, US: McGraw-Hill. pp. 251–270.
  7. ^ a b Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.