Extreme value theory

1 Tail limit theorems

The main result of use to our ends from EVT is the Pickands-Balkema-de Haan theorem (Balkema and de Haan 1974; Pickands III 1975).

This tells us that we can find a function $\beta (u)$ such that $$\underset{u\to t_T}{lim}\underset{0\le t<t_T-u}{sup}|T_u(t)-G_{\nu ,\beta (u),0}(t)|=0$$ if (and only if) $T$ is in the maximal domain of attraction of the extreme value distribution with parameter $\nu$ for some $\nu \in \mathbb{R}$.

This maximal domain of attraction was introduced in the Fisher-Tippett theorem (Fisher and Tippett 1928), and is analysed in the EVT literature (e.g. Embrechts, Kluppelberg, and Mikosch 1997). It is pretty hard to find a distribution that does not fit in the MDA. I should try.

Practically, this means that for many purposes, the tails of a random variable $\mathsf{t}$ may as well be assumed to be a GPD, $\mathsf{t}\sim G_{\nu ,\beta ,\mu }(t):=1-{(1+\frac{\nu (t-\mu )}{\beta })}^{-1/\nu }.$ Then for $t>s\ge 0$ and assuming that $G_{\nu ,\beta ,\mu }(s)>0,$ the survival probability over an interval $(s,t]$ is $$\begin{array}{rl}\mathbb{P}[\mathsf{t}\ge t\mid \mathsf{t}>s]& =\frac{\mathbb{P}[\mathsf{t}\ge t\cap \mathsf{t}>s]}{\mathbb{P}[\mathsf{t}>s]}\\ & =\frac{\mathbb{P}[\mathsf{t}\ge t]}{\mathbb{P}[\mathsf{t}>s]}\\ & =\frac{{\overline{G}}_{\nu ,\beta ,\mu }(t)}{{\overline{G}}_{\nu ,\beta ,\mu }(s)}\\ & =\frac{{(1+\frac{\nu (t-\mu )}{\beta })}^{-1/\nu }}{{(1+\frac{\nu (s-\mu )}{\beta })}^{-1/\nu }}\\ & ={\left(\frac{\beta +\nu (s-\mu )}{\beta +\nu (t-\mu )}\right)}^{1/\nu }.\end{array}$$

2 Generalized Pareto Distribution

Best intro from Hosking and Wallis (1987):

The generalized Pareto distribution is the distribution of a random variable $\mathsf{x}$ defined by $\mathsf{x}=\alpha (1-e^{-k\mathsf{y}})/k,$ where $\mathsf{y}$ is a random variable with the standard exponential distribution. The generalized Pareto distribution has distribution function

$$\begin{array}{rlrl}F(x)& =1-(1-kx/\alpha {)}^{1/k},& & k\ne 0\\ & =1-\exp (-x/\alpha ),& & k=0\end{array}$$ and density function

$$\begin{array}{rlrl}f(x)& ={\alpha }^{-1}(1-kx/\alpha {)}^{1/k-1},& & k\ne 0\\ & ={\alpha }^{-1}\exp (-x/\alpha ),& & k=0\end{array}$$ the range of $x$ is $0\le x<\mathrm{\infty }$ for $k\le 0$ and $0\le x\le \alpha /k$ for $k>0.$ The parameters of the distribution are $\alpha ,$ the scale parameter, and $k,$ the shape parameter. The special cases $k=0$ and $k=1$ yield, respectively, the exponential distribution with mean $\alpha$ and the uniform distribution on $[0,\alpha ];$ Pareto distributions are obtained when $k<0.$

and

1. The failure rate $r(x)=f(x)/\{1-F(x)\}$ is given by $r(x)=1/(\alpha -kx)$ and is monotonic in $x,$ decreasing if $k<0,$ constant if $k=0,$ and increasing if $k>0$
2. If the random variable $X$ has a generalized Pareto distribution, then the conditional distribution of $X-t$ given $X\ge t$ is also generalized Pareto, with the same value of $k$
3. Let $Z=max(0,X_1,\dots ,X_N),$ where the $X_i$ are independent and identically distributed as (1) and $N$ has a Poisson distribution. Then $Z$ has, essentially, a generalized extreme value (GEV) distribution as defined by Jenkinson (1955); that is, there exist quantities $\beta ,\gamma ,$ and $\delta ,$ independent of $z,$ such that

$$\begin{array}{rl}{F}_Z(z)& =\mathrm{Pr}(Z\le z)\\ & =\exp [-\{1-\delta (z-\gamma )/\beta {\}}^{1/\delta }],z\ge 0\end{array}$$ furthermore, $\delta =k;$ that is, the shape parameters of the GEV and the GPD are equal.

3 Generalized Extreme Value distributions

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4 Burr distribution

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5 References