Extreme Value Theory and Analysis
Nicholas Burke
12 February 2020
Extreme Value Theory
Outline
- What is EVT?
- Important Theorem
- Related Distributions
- Applications
Introduction
Extreme Value Theory is a section of statistics pertaining to extreme values. It is used to describes the limiting behavior of extremes \(max(X_2,...,X_n)\) or \(min(X_2,...,X_n)\), the tails of the distributions
Fisher-Tippet-Gnedenko Theorem
Let \(X_1,X_2,..,X_n\) be a sequence of independent and identically-distributed random variables and \(M_n=max({X_1,...,X_n})\). If as sequence of pairs of real numbers \((a_n,b_n)\) exists such that \(a_n>0\) and
\(\lim{P(\frac{M_n-b_n}{a_n})\leq x}=F(x)\)
where F is a non-degenerate distribution function, then the limit distribution F belongs to either the , the or the family, these can be grouped into the .
Generalized Extreme Value Distribution
Every extreme value distribution has a cumulative distribution function as follows
\(F(x:\sigma,\mu,\varepsilon)=e^{\{-[1+\varepsilon(\frac{x-\mu}{\sigma})]^{-\frac{1}{\varepsilon}}\}}\)
and probability distribution as follows
\(f(x:\sigma,\mu,\varepsilon)=(1+\varepsilon(\frac{x-\mu}{\sigma}))^{(\frac{-1}{\varepsilon})-1}e^{\{-[1+\varepsilon(\frac{x-\mu}{\sigma})]^{-\frac{1}{\varepsilon}}\}}\)
this known as the and defined by three parameters, parameter \(\mu\), the parameter \(\sigma\), and the parameter \(\varepsilon\). \(\sigma\) and \([1+\varepsilon(\frac{x-\mu}{\sigma})]\) must be greater than 0, \(\varepsilon\) and \(\mu\) can take on any real value.
GEVD
Gumbel
Minimum
\(F(x:\beta,\mu)=e^{x}e^{-e^{x}}\)
Maximum
\(F(x:\beta,\mu)=e^{-x}e^{-e^{x}}\)
where \(\mu,\beta\) are the and parameters, respectively
Gumbel
Fretchet
\(Pr(X\leq x)=e^{-\frac{x}{\alpha}}\)
Fretchet
Weibull
\(f(x)=\gamma x^{(\gamma-1)}e^{({-x-^{\gamma})}}\),
\(x \geq 0;\gamma >0\),
where \(\gamma,\alpha, \mu\) are the and parameters respectively
Weibull
Pickands-Balkema-deHaan Theorem
Let \((X_1,X_2,...)\) be a sequence of independent and identically distribution random variables and let \(F_u\) be their conditional excess distribution function. For a large class of underlying distribution functions \(F\), and large \(u\), \(F_u\) is well approximated by the . That is
\(F_u(y)=G_{k,\sigma}(y)\) as \(u\rightarrow \infty\)
where
\[ G_{k,\sigma}(y)=\left\{ \begin{array}{ll} 1-(1+\frac{ky}{\sigma})^{\frac{-1}{k}},&\text{if k $\neq$ 0}\\1-e^{\frac{-y}{\sigma}},&\text {if k = 0} \end{array} \right. \]
Generalized Pareto Distribution
The standard cumulative distribution function is as follows
\[ F_{\varepsilon}(z)=\left\{ \begin{array}{ll} 1-(1+\varepsilon z)^{\frac{-1}{\varepsilon}},&\text{for $\varepsilon \neq 0$}\\ 1-e^{-z},&\text {for $\varepsilon =0$} \end{array} \right. \]
where \(\varepsilon\) is the parameter
- when \(z \geq 0\) for \(\varepsilon \geq 0\)
- when \(0\leq z \leq \frac{-1}{\varepsilon}\) for \(\varepsilon <0\)
GPD
Pareto
\(F(x)=1-x^{-\alpha}\), \(x\geq 1,\alpha >0\)
Pareto
Exponential
\(F(x)=1-e^{-x}\), \(x>0\) ## Exponential 
Beta
\(F(x)=1-(-x)^{-\alpha}\); \(-1\geq x\geq 0,\alpha <0\)
Beta
Extreme Value Analysis
Extreme Value Analysis is essentially the estimation of the shape, location and scale parameters
Block Maxima
If appropriately normalized block maxima converge in distribution to a non-trivial limit distribution, then this distribution is the generalized extreme value (GEV) distribution as follows;
\[ H_{\varepsilon}(x)=\left\{ \begin{array}{ll} e^{-(1+\varepsilon x)^{\frac{-1}{\varepsilon}}},&\text{for $\varepsilon \neq 0$}\\ e^{-e^{-x}},&\text {for $\varepsilon =0$} \end{array} \right. \]
where \(\varepsilon\) is the parameter, that determine whether it belongs to or family of distributions .
Block Maxima
Block Maxima
Peak-over-Threshold
For a class of distributions a function \(\beta(u)\) can be found such that
\(\lim_{u\rightarrow \bar{x}} \sup_{0\leq x < \bar{x}-u}|F_u(x)-G_{\varepsilon, \beta(u)}|=0\)
where, \(\bar{x}\) is the rightmost point of the distribution and \(u\) is the threshold point
Peak-over-Threshold
Peak-over-Threshold
