Instatistics, theFisher–Tippett–Gnedenko theorem (also theFisher–Tippett theorem or theextreme value theorem) is a general result inextreme value theory regarding asymptotic distribution of extremeorder statistics. The maximum of a sample ofiidrandom variables after proper renormalization can onlyconverge in distribution to one of three possibledistribution families: theGumbel distribution, theFréchet distribution, or theWeibull distribution. Credit for the extreme value theorem and its convergence details are given toFréchet (1927),^[1]Fisher andTippett (1928),^[2]von Mises (1936),^[3][4] andGnedenko (1943).^[5]

The role of the extremal types theorem for maxima is similar to that ofcentral 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 thatif 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${\displaystyle X_1,X_2,\dots ,X_n}$ be ann-sized sample ofindependent and identically-distributed random variables, each of whosecumulative distribution function is${\displaystyle F}$. 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:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mathbb{P}\left(\frac{max\{X_1,\dots ,X_n\}-b_n}{a_n}\le x\right)=G(x),}$

or equivalently:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(F(a_{n}x+b_n){)}^n=G(x).}$

In such circumstances, the limiting function${\displaystyle G}$ is the cumulative distribution function of a distribution belonging to either theGumbel, theFréchet, or theWeibulldistribution family.^[6]

In other words, if the limit above converges, then up to a linear change of coordinates${\displaystyle G(x)}$ will assume either the form:^[7]

${\displaystyle G_{\gamma }(x)=\exp (-(1+\gamma x{)}^{-1/\gamma })\text{for }\gamma \ne 0,}$

with the non-zero parameter${\displaystyle \gamma }$ also satisfying${\displaystyle 1+\gamma x>0}$ for every${\displaystyle x}$ value supported by${\displaystyle F}$ (for all values${\displaystyle x}$ for which${\displaystyle F(x)\ne 0}$).^{[clarification needed]} Otherwise it has the form:

${\displaystyle G_0(x)=\exp (-\exp (-x))\text{for }\gamma =0.}$

This is the cumulative distribution function of thegeneralized extreme value distribution (GEV) with extreme value index${\displaystyle \gamma }$. The GEV distribution groups the Gumbel, Fréchet, and Weibull distributions into a single composite form.

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 (1936)^[3][5][4] and was further developed by Gnedenko (1943).^[5]

Let${\displaystyle F}$ be the distribution function of${\displaystyle X}$, and${\displaystyle X_1,\dots ,X_n}$ be somei.i.d. sample thereof. Also let${\displaystyle x_{\mathsf{m}\mathsf{a}\mathsf{x}}}$ be the population maximum:.

Then the limiting distribution of the normalized sample maximum, given by${\displaystyle G}$ above, will then be one of the following three types:^[7]

• Fréchet distribution (${\displaystyle \gamma >0}$): For strictly positive${\displaystyle \gamma >0}$, the limiting distribution converges if and only if${\displaystyle x_{\mathsf{m}\mathsf{a}\mathsf{x}}=\mathrm{\infty }}$ and

${\displaystyle \underset{t\to \mathrm{\infty }}{lim}\frac{1-F(ut)}{1-F(t)}=u^{1/\gamma }}$ for all${\displaystyle u>0}$.

In this case, possible sequences that will satisfy the theorem conditions are${\displaystyle b_n=0}$ and${\displaystyle a_n=F^{-1}\left(1-\frac{1}{n}\right)}$. Strictly positive${\displaystyle \gamma }$ corresponds to what is called aheavy tailed distribution.

• Gumbel distribution (${\displaystyle \gamma =0}$): For trivial${\displaystyle \gamma =0}$, and with${\displaystyle x_{\mathsf{m}\mathsf{a}\mathsf{x}}}$ either finite or infinite, the limiting distribution converges if and only if

${\displaystyle \underset{t\to x_{\mathsf{m}\mathsf{a}\mathsf{x}}}{lim}\frac{1-F(t+u\tilde{g}(t))}{1-F(t)}={\mathrm{e}}^{-u}}$ for all${\displaystyle u>0}$ with${\displaystyle \tilde{g}(t)\equiv \frac{{\int }_t^{x_{\mathsf{m}\mathsf{a}\mathsf{x}}}(1-F(s))\mathrm{d}s}{1-F(t)}}$.

Possible sequences here are${\displaystyle b_n=F^{-1}\left(1-\frac{1}{n}\right)}$ and${\displaystyle a_n=\tilde{g}(F^{-1}\left(1-\frac{1}{n}\right))}$.

• Weibull distribution (): For strictly negative, the limiting distribution converges if and only if (is finite) and

${\displaystyle \underset{t\to 0^{+}}{lim}\frac{1-F(x_{\mathsf{m}\mathsf{a}\mathsf{x}}-ut)}{1-F(x_{\mathsf{m}\mathsf{a}\mathsf{x}}-t)}=u^{-1/\gamma }}$ for all${\displaystyle u>0}$.

Note that for this case the exponential term${\displaystyle -1/\gamma }$ is strictly positive, since${\displaystyle \gamma }$ is strictly negative.

Possible sequences here are${\displaystyle b_n=x_{\mathsf{m}\mathsf{a}\mathsf{x}}}$ and${\displaystyle a_n=x_{\mathsf{m}\mathsf{a}\mathsf{x}}-F^{-1}\left(1-\frac{1}{n}\right)}$.

Note that the second formula (the Gumbel distribution) is the limit of the first (the Fréchet distribution) as${\displaystyle \gamma }$ goes to zero.

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TheCauchy distribution's density function is:

${\displaystyle f(x)=\frac{1}{{\pi }^2+x^2},}$

and its cumulative distribution function is:

${\displaystyle F(x)=\frac{1}{2}+\frac{1}{\pi }\arctan \left(\frac{x}{\pi }\right).}$

A little bit of calculus show that the right tail's cumulative distribution${\displaystyle 1-F(x)}$ isasymptotic to${\displaystyle \frac{1}{x},}$ or

${\displaystyle \ln F(x)\to \frac{-1}{x}\mathsf{a}\mathsf{s}x\to \mathrm{\infty },}$

so we have

${\displaystyle \ln \left(F(x{)}^n\right)=n\ln F(x)\sim -\frac{-n}{x}.}$

Thus we have

${\displaystyle F(x{)}^n\approx \exp \left(\frac{-n}{x}\right)}$

and letting${\displaystyle u\equiv \frac{x}{n}-1}$ (and skipping some explanation)

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(F(nu+n{)}^n)=\exp \left(\frac{-1}{1+u}\right)=G_1(u)}$

for any${\displaystyle u.}$

Let us take thenormal distribution with cumulative distribution function

${\displaystyle F(x)=\frac{1}{2}\mathrm{erfc}\left(\frac{-x}{\sqrt{2}}\right).}$

We have

${\displaystyle \ln F(x)\to -\frac{\exp \left(-\frac{1}{2}{x}^2\right)}{\sqrt{2\pi }x}\mathsf{a}\mathsf{s}x\to \mathrm{\infty }}$

and thus

${\displaystyle \ln \left(F(x{)}^n\right)=n\ln F(x)\to -\frac{n\exp \left(-\frac{1}{2}{x}^2\right)}{\sqrt{2\pi }x}\mathsf{a}\mathsf{s}x\to \mathrm{\infty }.}$

Hence we have

${\displaystyle F(x{)}^n\approx \exp \left(-\frac{n\exp \left(-\frac{1}{2}{x}^2\right)}{\sqrt{2\pi }x}\right).}$

If we define${\displaystyle c_n}$ as the value that exactly satisfies

${\displaystyle \frac{n\exp \left(-\frac{1}{2}{c}_n^2\right)}{\sqrt{2\pi }{c}_n}=1,}$

then around${\displaystyle x=c_n}$

${\displaystyle \frac{n\exp \left(-\frac{1}{2}{x}^2\right)}{\sqrt{2\pi }x}\approx \exp \left(c_n(c_n-x)\right).}$

As${\displaystyle n}$ increases, this becomes a good approximation for a wider and wider range of${\displaystyle c_n(c_n-x)}$ so letting${\displaystyle u\equiv c_n(x-c_n)}$ we find that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(F{\left(\frac{u}{c_n}+c_n\right)}^n)=\exp (-\exp (-u))=G_0(u).}$

Equivalently,

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mathbb{P}(\frac{max\{X_1,\dots ,X_n\}-c_n}{\left(\frac{1}{c_n}\right)}\le u)=\exp (-\exp (-u))=G_0(u).}$

With this result, we see retrospectively that we need${\displaystyle \ln c_n\approx \frac{\ln \ln n}{2}}$ and then

${\displaystyle c_n\approx \sqrt{2\ln n},}$

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

We may take the simplest example, a uniform distribution between0 and1, with cumulative distribution function

${\displaystyle F(x)=x}$ for anyx value from0 to1 .

For values of${\displaystyle x\to 1}$ we have

${\displaystyle \ln (F(x{)}^n)=n\ln F(x)\to n(1-x).}$

So for${\displaystyle x\approx 1}$ we have

${\displaystyle F(x{)}^n\approx \exp (n-nx).}$

Let${\displaystyle u\equiv 1+n(1-x)}$ and get

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(F\left(\frac{u}{n}+1-\frac{1}{n}\right){)}^n=\exp (-(1-u))=G_{-1}(u).}$

Close examination of that limit shows that the expected maximum approaches1 in inverse proportion ton .

• Lee, Seyoon; Kim, Joseph H.T. (8 March 2018)."Exponentiated generalized Pareto distribution".Communications in Statistics – Theory and Methods.48 (8) (online ed.):2014–2038.arXiv:1708.01686.doi:10.1080/03610926.2018.1441418.ISSN 1532-415X – via tandfonline.com.

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