# Fisher–Tippett–Gnedenko 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 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.

Contents 1 Statement 2 Conditions of convergence 3 Examples 3.1 Fréchet distribution 3.2 Gumbel distribution 3.3 Weibull distribution 4 See also 5 Notes Statement Let {displaystyle X_{1},X_{2},ldots ,X_{n}} be a sequence of independent and identically-distributed random variables with cumulative distribution function {displaystyle F} . Suppose that there exist two sequences of real numbers {displaystyle a_{n}>0} and {displaystyle b_{n}in mathbb {R} } such that the following limits converge to a non-degenerate distribution function: {displaystyle lim _{nto infty }Pleft({frac {max{X_{1},dots ,X_{n}}-b_{n}}{a_{n}}}leq xright)=G(x)} , or equivalently: {displaystyle lim _{nto infty }F^{n}left(a_{n}x+b_{n}right)=G(x)} .

In such circumstances, the limit distribution {displaystyle G} belongs to either the Gumbel, the Fréchet or the Weibull family.[6] In other words, if the limit above converges we will have {displaystyle G(x)} assume the form:[7] {displaystyle G_{gamma }(x)=exp left(-(1+gamma ,x)^{-1/gamma }right),;;1+gamma ,x_{a,b}>0} or else {displaystyle G_{0}(x)=exp left(-exp(-x)right)} for some parameter {displaystyle gamma .} This is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index {displaystyle gamma } . 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 {displaystyle gamma } goes to zero.

Conditions of convergence The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution {displaystyle G(x)} above. The study of conditions for convergence of {displaystyle G} 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 {displaystyle F} be the distribution function of {displaystyle X} , and {displaystyle X_{1},dots ,X_{n}} an i.i.d. sample thereof. Also let {displaystyle x^{*}} be the populational maximum, i.e. {displaystyle x^{*}=sup{xmid F(x)<1}} . The limiting distribution of the normalized sample maximum, given by {displaystyle G} above, will then be:[7] A Fréchet distribution ( {displaystyle gamma >0} ) if and only if {displaystyle x^{*}=infty } and {displaystyle lim _{trightarrow infty }{frac {1-F(ut)}{1-F(t)}}=u^{-1/|gamma |}} for all {displaystyle u>0} . This corresponds to what we may call a heavy tail. In this case, possible sequences that will satisfy the theorem conditions are {displaystyle b_{n}=0} and {displaystyle a_{n}=F^{-1}left(1-{frac {1}{n}}right)} . A Gumbel distribution ( {displaystyle gamma =0} ), with {displaystyle x^{*}} finite or infinite, if and only if {displaystyle lim _{trightarrow x^{*}}{frac {1-F(t+uf(t))}{1-F(t)}}=e^{-u}} for all {displaystyle u>0} with {displaystyle f(t):={frac {int _{t}^{x^{*}}1-F(s)ds}{1-F(t)}}} . Possible sequences here are {displaystyle b_{n}=F^{-1}left(1-{frac {1}{n}}right)} and {displaystyle a_{n}=fleft(F^{-1}left(1-{frac {1}{n}}right)right)} . A Weibull distribution ( {displaystyle gamma <0} ) if and only if {displaystyle x^{*}} is finite and {displaystyle lim _{trightarrow 0^{+}}{frac {1-F(x^{*}-ut)}{1-F(x^{*}-t)}}=u^{1/|gamma |}} for all {displaystyle u>0} . Possible sequences here are {displaystyle b_{n}=x^{*}} and {displaystyle a_{n}=x^{*}-F^{-1}left(1-{frac {1}{n}}right)} . Examples Fréchet distribution If we take the Cauchy distribution {displaystyle f(x)=(pi ^{2}+x^{2})^{-1}} the cumulative distribution function is: {displaystyle F(x)=1/2+{frac {1}{pi }}arctan(x/pi )} {displaystyle 1-F(x)} is asymptotic to {displaystyle 1/x,} or {displaystyle ln F(x)sim -1/x} and we have {displaystyle ln F(x)^{n}=nln F(x)sim -n/x.} Thus we have {displaystyle F(x)^{n}approx exp(-n/x)} and letting {displaystyle u=x/n-1} (and skipping some explanation) {displaystyle lim _{nto infty }F(nu+n)^{n}=exp(-(1+u)^{-1})=G_{1}(u)} for any {displaystyle u.} The expected maximum value therefore goes up linearly with n.

Gumbel distribution Let us take the normal distribution with cumulative distribution function {displaystyle F(x)={frac {1}{2}}{text{erfc}}(-x/{sqrt {2}}).} We have {displaystyle ln F(x)sim -{frac {exp(-x^{2}/2)}{{sqrt {2pi }}x}}} and {displaystyle ln F(x)^{n}=nln F(x)sim -{frac {nexp(-x^{2}/2)}{{sqrt {2pi }}x}}.} Thus we have {displaystyle F(x)^{n}approx exp left(-{frac {nexp(-x^{2}/2)}{{sqrt {2pi }}x}}right).} If we define {displaystyle c_{n}} as the value that satisfies {displaystyle {frac {nexp(-c_{n}^{2}/2)}{{sqrt {2pi }}c_{n}}}=1} then around {displaystyle x=c_{n}} {displaystyle {frac {nexp(-x^{2}/2)}{{sqrt {2pi }}x}}approx exp(c_{n}(c_{n}-x)).} As n increases, this becomes a good approximation for a wider and wider range of {displaystyle c_{n}(c_{n}-x)} so letting {displaystyle u=c_{n}(c_{n}-x)} we find that {displaystyle lim _{nto infty }F(u/c_{n}+c_{n})^{n}=exp(-exp(-u))=G_{0}(u).} Equivalently, {displaystyle lim _{nto infty }P{Bigl (}{frac {max{X_{1},ldots ,X_{n}}-c_{n}}{1/c_{n}}}leq u{Bigr )}=exp(-exp(-u))=G_{0}(u).} We can see that {displaystyle ln c_{n}sim (ln ln n)/2} and then {displaystyle c_{n}sim {sqrt {2ln n}}} so the maximum is expected to climb ever more slowly toward infinity.

Weibull distribution We may take the simplest example, a uniform distribution between 0 and 1, with cumulative distribution function {displaystyle F(x)=x} from 0 to 1.

Approaching 1 we have {displaystyle ln F(x)^{n}=nln F(x)sim -n(1-x).} Then {displaystyle F(x)^{n}approx exp(nx-n).} Letting {displaystyle u=1+n(1-x)} we have {displaystyle lim _{nto infty }F(u/n+1)^{n}=exp left(-(1-u)right)=G_{-1}(u).} The expected maximum approaches 1 inversely proportionally to n.

See also Extreme value theory Gumbel distribution Generalized extreme value distribution Pickands–Balkema–de Haan theorem Generalized Pareto distribution Exponentiated generalized Pareto distribution Notes ^ Fréchet, M. (1927), "Sur la loi de probabilité de l'écart maximum", Annales de la Société Polonaise de Mathématique, 6 (1): 93–116 ^ 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 ^ Jump up to: a b Mises, R. von (1936). "La distribution de la plus grande de n valeurs". Rev. Math. Union Interbalcanique 1: 141–160. ^ Jump up to: a b Falk, Michael; Marohn, Frank (1993). "Von Mises conditions revisited". The Annals of Probability: 1310–1328. ^ Jump up to: 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 ^ Mood, A.M. (1950). "5. Order Statistics". Introduction to the theory of statistics. New York, NY, US: McGraw-Hill. pp. 251–270. ^ Jump up to: a b Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer. Categories: Theorems in statisticsExtreme value dataTails of probability distributions

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