Extreme Values#

Part of a series: Stochastic fundamentals.

Follow reading here

Consider (again) a sequence of iid random variables \(X_1, X_2, ...\). Extreme value theory analyzes the probabiltiy of finding a certain [maximum]{.ul}

\(\begin{aligned} M_n &= \max (X_1, X_2, ..., X_n) \end{aligned}\)

after \(n\) repetitions.

Why is this important?

  • If you want to build a nuclear power plant at the Japanese Eastern Coast, you have to know which maximum flood you should expect during a long time span (\(> 100\) years) at this point. The Tsunami waves at Fukushima on 11.3.2011 reached \(> 13\)m, while the seawall was lower than \(10\)m.

  • Can the observed summer heat waves (e.g., 27 July 2013 in Germany) be attributed to Climate Change or are they just natural extreme events?
    \(\rightarrow\) See [Wergen et al., 2014]

Let us try to calculate the CDF for \(M_n\). We have

\(\begin{aligned} P(M_n \leq z) &= P(X_1 \leq z, X_2 \leq z, \dots, X_n \leq z)\\ &= P(X_1 \leq z) P(X_1 \leq z) \dots P(X_1 \leq z)\\ &= \left[ F(z) \right]^n \end{aligned}\)

What can we say about the limit of large n? Obviously the maximum will increase with n so we have to do some rescaling. The question is thus: Can we find a CDF \(G(z)\) such that $\(\begin{aligned} P((M_N - b_n) | a_n \leq z) \xrightarrow{n \rightarrow \infty} G(z)\end{aligned}\)\( for some sequence of numbers \)a_n, b_n\( with \)a_n > 0$?

Example: Exponential Distribution#

\(\begin{aligned} f_x(\kappa) &= \left\{\begin{array}{ l l } \lambda \cdot e^{- \lambda \kappa} & \kappa \geq 0 \\ 0 & \kappa < 0 \end{array}\right. \end{aligned}\)

\(\begin{aligned} F_x(\kappa) &= \left\{\begin{array}{ l l } 1 - e^{- \lambda \kappa} & \kappa \geq 0 \\ 0 & \kappa < 0 \end{array}\right. \end{aligned}\)

Can we find \(a_n, b_n, G(\kappa)\) such that

\(\begin{aligned} F^n (a_n \kappa + b_n) &\xrightarrow{n \rightarrow \infty} G(\kappa)\\ \left( 1- e^{-\lambda(a_n \kappa + b_n)} \right)^n &\xrightarrow{n \rightarrow \infty} G(\kappa)? \end{aligned}\)

Take the log

\(\begin{aligned} n \cdot \ln{\left( 1 - e^{- \lambda (a_n \kappa + b_n)} \right)} &\xrightarrow{n \rightarrow \infty} \ln G(\kappa)? \end{aligned}\)

Taylorexpansion for the logarithm

\(\begin{aligned} n e^{-a_n \kappa - b_n} &\xrightarrow{n \rightarrow \infty} - ln G(\kappa)? \end{aligned}\)

We can make the left-hand side n-independent by choosing

\(\begin{aligned} b_n &= \ln n \Leftrightarrow n e^{-b_n} = 1 \end{aligned}\)


\(\begin{aligned} e^{-a_n \kappa} &\xrightarrow{n \rightarrow \infty} - \ln G(\kappa)\end{aligned}\)

simply setting \(a_n = 1\) then yields

\(\begin{aligned} G(\kappa) &= e^{-e^{-\kappa}} \end{aligned}\)

Theorem (Fischer- Tippet-Gnedenko) Let \(X_1, X_2, \dots, X_n\) be a sequence of iid random variables and \(M_n = max(X_1, X_2, \dots, X_n)\). If a sequence of real number \((a_n, b_n)\) with \(a_n > 0\) existis such that

\(\begin{aligned} \lim_{n\to\infty} P\left( \frac{M_n-b_n}{a_n} \leq z \right) &= G(z) \end{aligned}\)

with a non-degenerate CDF \(G(z)\), then \(G(z)\) does not belong to one of the following three families:

  1. Weibull (type 3): \(\begin{aligned} G(z) &= \left\{ \begin{array}{ll} e^{-(-z)^\alpha} & z < 0 \\ 1 & z \geq 0 \end{array} \right. \end{aligned}\)

  2. Gumbel: \(\begin{aligned} G(z) &= \begin{array}{ll} e^{-e^{-z}} & z \in \mathbb{R} \end{array} \end{aligned}\)

  3. Fréchet: \(\begin{aligned} G(z) &= \left\{ \begin{array}{ll} 0 & z \leq 0 \\ e^{-z^{-\alpha}} & z > 0 \end{array} \right. \end{aligned}\)



Gregor Wergen, Joachim Krug, and Stefan Rahmstadt. 2014. URL: https://www.spektrum.de/magazin/klimarekorde/1216444#.


Philipp Böttcher, Dirk Witthaut