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 only 3 possible distribution families: 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] Fisher and 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

Let ${\displaystyle \ X_{1},X_{2},\ldots ,X_{n}\ }$ be an n-sized sample of independent and identically-distributed random variables, each of whose cumulative 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:

${\displaystyle \lim _{n\to \infty }{\boldsymbol {\mathcal {P}}}\left\{{\frac {\ \max\{X_{1},\dots ,X_{n}\}-b_{n}\ }{a_{n}}}\leq x\ \right\}=G(x)\ ,}$

or equivalently:

${\displaystyle \lim _{n\to \infty }{\Bigl (}\ F\left(\ a_{n}\ x+b_{n}\ \right){\Bigr )}^{n}=G(x)~.}$

In such circumstances, the limiting distribution ${\displaystyle \ G\ }$ belongs to either the Gumbel, the Fréchet, or the Weibull distribution 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]

${\displaystyle G_{\gamma }(x)=\exp \left(-{\Bigl (}1+\gamma \ x{\Bigr )}^{\left({\tfrac {-1\;}{\gamma }}\right)}\right)\quad }$ for ${\displaystyle \quad \gamma \neq 0\ ,}$

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 ${\displaystyle \ F(x)\neq 0\ }$). Otherwise it has the form:

${\displaystyle G_{0}(x)=\exp {\bigl (}\ -\exp(-x)\ {\bigr )}\quad }$ for ${\displaystyle \quad \gamma =0~.}$

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 composite form.

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 (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 some i.i.d. sample thereof.

Also let ${\displaystyle \ x_{\mathsf {max}}\ }$ be the population maximum: ${\displaystyle \ x_{\mathsf {max}}\equiv \sup \ \{\ x\ \mid \ F(x)<1\ \}~.\ }$

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

Fréchet distribution ${\displaystyle \ \left(\ \gamma >0\ \right)}$

For strictly positive ${\displaystyle \ \gamma >0\ ,}$ the limiting distribution converges if and only if

${\displaystyle \ x_{\mathsf {max}}=\infty \ }$

and

${\displaystyle \ \lim _{t\rightarrow \infty }{\frac {\ 1-F(u\ t)\ }{1-F(t)}}=u^{\left({\tfrac {-1~}{\gamma }}\right)}\ }$ for all ${\displaystyle \ u>0~.}$

In this case, possible sequences that will satisfy the theorem conditions are

${\displaystyle b_{n}=0}$

and

${\displaystyle \ a_{n}={F^{-1}}\!\!\left(1-{\tfrac {1}{\ n\ }}\right)~.}$

Strictly positive ${\displaystyle \ \gamma \ }$ corresponds to what is called a heavy tailed distribution.

Gumbel distribution ${\displaystyle \ \left(\ \gamma =0\ \right)}$

For trivial ${\displaystyle \ \gamma =0\ ,}$ and with ${\displaystyle \ x_{\mathsf {max}}\ }$ either finite or infinite, the limiting distribution converges if and only if

${\displaystyle \ \lim _{t\rightarrow x_{\mathsf {max}}}{\frac {\ 1-F{\bigl (}\ t+u\ {\tilde {g}}(t)\ {\bigr )}\ }{1-F(t)}}=e^{-u}\ }$ for all ${\displaystyle \ u>0\ }$

with

${\displaystyle \ {\tilde {g}}(t)\equiv {\frac {\ \int _{t}^{x_{\mathsf {max}}}{\Bigl (}\ 1-F(s)\ {\Bigr )}\ \mathrm {d} \ s\ }{1-F(t)}}~.}$

Possible sequences here are

${\displaystyle \ b_{n}={F^{-1}}\!\!\left(\ 1-{\tfrac {1}{\ n\ }}\ \right)\ }$

and

${\displaystyle \ a_{n}={\tilde {g}}{\Bigl (}\;{F^{-1}}\!\!\left(\ 1-{\tfrac {1}{\ n\ }}\ \right)\;{\Bigr )}~.}$

Weibull distribution ${\displaystyle \ \left(\ \gamma <0\ \right)}$

For strictly negative ${\displaystyle \ \gamma <0\ }$ the limiting distribution converges if and only if

${\displaystyle \ x_{\mathsf {max}}\ <\infty \quad }$ (is finite)

and

${\displaystyle \ \lim _{t\rightarrow 0^{+}}{\frac {\ 1-F\!\left(\ x_{\mathsf {max}}-u\ t\ \right)\ }{1-F(\ x_{\mathsf {max}}-t\ )}}=u^{\left({\tfrac {-1~}{\ \gamma \ }}\right)}\ }$ for all ${\displaystyle \ u>0~.}$

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

Possible sequences here are

${\displaystyle \ b_{n}=x_{\mathsf {max}}\ }$

and

${\displaystyle \ a_{n}=x_{\mathsf {max}}-{F^{-1}}\!\!\left(\ 1-{\frac {1}{\ n\ }}\ \right)~.}$

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

Examples

Fréchet distribution

The Cauchy 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 ${\displaystyle \ 1-F(x)\ }$ is asymptotic to ${\displaystyle \ {\frac {1}{\ x\ }}\ ,}$ or

${\displaystyle \ln F(x)\rightarrow {\frac {-1~}{\ x\ }}\quad {\mathsf {~as~}}\quad x\rightarrow \infty \ ,}$

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)

${\displaystyle \lim _{n\to \infty }{\Bigl (}\ F(n\ u+n)^{n}\ {\Bigr )}=\exp \left({\tfrac {-1~}{\ 1+u\ }}\right)=G_{1}(u)\ }$

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

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

We have

${\displaystyle \ln F(x)\rightarrow -{\frac {\ \exp \left(-{\tfrac {1}{2}}x^{2}\right)\ }{{\sqrt {2\pi \ }}\ x}}\quad {\mathsf {~as~}}\quad x\rightarrow \infty }$

and thus

${\displaystyle \ln \left(\ F(x)^{n}\ \right)=n\ln F(x)\rightarrow -{\frac {\ n\exp \left(-{\tfrac {1}{2}}x^{2}\right)\ }{{\sqrt {2\pi \ }}\ x}}\quad {\mathsf {~as~}}\quad x\rightarrow \infty ~.}$

Hence we have

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

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

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

then around ${\displaystyle \ x=c_{n}\ }$

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

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

${\displaystyle \lim _{n\to \infty }{\biggl (}\ F\left({\tfrac {u}{~c_{n}\ }}+c_{n}\right)^{n}\ {\biggr )}=\exp \!{\Bigl (}-\exp(-u){\Bigr )}=G_{0}(u)~.}$

Equivalently,

${\displaystyle \lim _{n\to \infty }{\boldsymbol {\mathcal {P}}}\ {\Biggl (}{\frac {\ \max\{X_{1},\ \ldots ,\ X_{n}\}-c_{n}\ }{\left({\frac {u}{~c_{n}\ }}\right)}}\leq u{\Biggr )}=\exp \!{\Bigl (}-\exp(-u){\Bigr )}=G_{0}(u)~.}$

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.

Weibull distribution

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

${\displaystyle F(x)=x\ }$ for any x value from 0 to 1 .

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

${\displaystyle \ln {\Bigl (}\ F(x)^{n}\ {\Bigr )}=n\ \ln F(x)\ \rightarrow \ n\ (\ 1-x\ )~.}$

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

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

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

${\displaystyle \lim _{n\to \infty }{\Bigl (}\ F\!\left({\tfrac {\ u\ }{n}}+1-{\tfrac {\ 1\ }{n}}\right)\ {\Bigr )}^{n}=\exp \!{\bigl (}\ -(1-u)\ {\bigr )}=G_{-1}(u)~.}$

Close examination of that limit shows that the expected maximum approaches 1 in inverse proportion to n .

See also

References

Further reading

• 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.