Extreme Values

Extreme Values#

Part of a series: Stochastic fundamentals.

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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

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

Then

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

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:

Weibull (type 3): $\begin{aligned} G(z) &= \left\{ \begin{array}{ll} e^{-(-z)^\alpha} & z < 0 \\ 1 & z \geq 0 \end{array} \right. \end{aligned}$
Gumbel: $\begin{aligned} G(z) &= \begin{array}{ll} e^{-e^{-z}} & z \in \mathbb{R} \end{array} \end{aligned}$
Fréchet: $\begin{aligned} G(z) &= \left\{ \begin{array}{ll} 0 & z \leq 0 \\ e^{-z^{-\alpha}} & z > 0 \end{array} \right. \end{aligned}$

References#

WKR14: Gregor Wergen, Joachim Krug, and Stefan Rahmstadt. 2014. URL: https://www.spektrum.de/magazin/klimarekorde/1216444#.

Authors#

Philipp Böttcher, Dirk Witthaut