Extreme value theory

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 \[\lim _{u \rightarrow t_{T}} \sup _{0 \leq t<t_{T}-u}\left|T_{u}(t)-G_{\nu, \beta(u),0}(t)\right|=0\] if (and only if) \(T\) is in the maximal domain of attraction of the extreme value distribution with parameter \(\nu\) for some \(\nu\in\bb{R}\).

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 ( may as well be assumed to be a GPD, \(\rv{t}\sim G_{\nu,\beta,\mu}(t) := 1-\left(1+\frac{\nu (t-\mu)}{\beta}\right)^{-1/\nu}.\) Then for \(t> s\geq 0\) and assuming that \(G_{\nu,\beta,\mu}(s)>0,\) the survival probability over an interval \((s,t]\) is \[\begin{aligned} \Pr[\rv{t}\geq t\gvn \rv{t}> s] &=\frac{\Pr[\rv{t}\geq t\cap \rv{t}> s]}{\Pr[\rv{t}> s]}\\ &=\frac{\Pr[\rv{t}\geq t]}{\Pr[\rv{t}> s]}\\ &=\frac{\bar{G}_{\nu,\beta,\mu}(t)}{\bar{G}_{\nu,\beta,\mu}(s)}\\ &=\frac{\left(1+\frac{\nu (t-\mu)}{\beta}\right)^{-1/\nu}}{\left(1+\frac{\nu (s-\mu)}{\beta}\right)^{-1/\nu}}\\ &=\left(\frac{\beta+\nu (s-\mu)}{\beta+\nu (t-\mu)}\right)^{1/\nu}.\label{eq:gpd-survival-prob} \end{aligned}\]

Generalized Pareto Distribution

Best intro from Hosking and Wallis (1987):

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

\[ \begin{aligned} F(x) &=1-(1-k x / \alpha)^{1 / k}, & & k \neq 0 \\ &=1-\exp (-x / \alpha), & & k=0 \end{aligned} \] and density function

\[ \begin{aligned} f(x) &=\alpha^{-1}(1-k x / \alpha)^{1 / k-1}, & & k \neq 0 \\ &=\alpha^{-1} \exp (-x / \alpha), & & k=0 \end{aligned} \] the range of \(x\) is \(0 \leq x<\infty\) for \(k \leq 0\) and \(0 \leq x \leq \alpha / k\) 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 \(r(x)=1 /(\alpha-k x)\) 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 \(X \geq t\) is also generalized Pareto, with the same value of \(k\)
3. Let \(Z=\max \left(0, X_{1}, \ldots, X_{N}\right),\) 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

\[ \begin{aligned} F_{Z}(z) &=\operatorname{Pr}(Z \leq z) \\ &=\exp \left[-\{1-\delta(z-\gamma) / \beta\}^{1 / \delta}\right], \quad z \geq 0 \end{aligned} \] furthermore, \(\delta=k ;\) that is, the shape parameters of the GEV and the GPD are equal.

Generalized Extreme Value distributions

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Burr distribution

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References