Written on April 17, 2021.

Tags: statistics, cool stuffs, extreme values

Table of contents

• Pickands–Balkema–De Haan theorem
• Peaks-Over-Threshold (POT) method
• Next
• References

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Previous post covered the first theorem of EVT and Block Maxima and I pointed that a common grief on this method is that it is not using ‘all the data’. This post will focus on the ‘second theorem’ of EVT aka Pickands–Balkema–De Haan theorem.

Pickands–Balkema–De Haan theorem

Let’s start with the theorem.

Let \({X}\) be an r.v. of unknown distribution function \(F\) with right endpoint \(x_F \coloneqq sup\{x : F(x) < 1\} \leq \infty\) and for which the first theorem of EVT is satisfied¹. Pickands–Balkema–De Haan theorem says that the conditional excess distribution over a threshold \(u\):

\[ F_u(y) \coloneqq Pr(X - u \leq y | X > u) = \frac{F(u + y) - F(u)}{1 - F(u)} \]

for \(0 \leq y \leq x_F - u\) is well approximated by the Generalized Pareto Distribution (GPD).

That is:

\[F_{u}(y)\rightarrow G_{{\xi,\sigma }}(y),{\text{ as }}u\rightarrow \infty\]

where

\[ G_{{\xi,\sigma }}(y) \coloneqq \begin{cases} 1 - \bigg( 1 + \frac{\xi y}{\sigma} \bigg)^{-1/\xi}& \text{if } \xi \neq 0\\ 1 - e^{-y/\sigma} & \text{otherwise.} \end{cases} \]

In plain text now: if we look asymptotically at the values above a certain threshold, they seem to be drawn from a GPD distribution characterized by only 2 parameters – there is only a limited number of ways a tale of tails ends.

\(\xi\) is called the shape parameter or tail index and as its name suggests define the shape of the tail and how heavy the tail is. \(\sigma\) is the scale parameter. There is actually a relation between these parameters and the one of the GEV of the first theorem, check An Introduction to Statistical Modeling of Extremes Values, Stuart Coles if you want it.

One note: this is for \(u\rightarrow \infty\) so when one gets to choose \(u\), \(u\) should be taken ‘large enough’ - some high percentile for example or using graphical approaches based on Mean Residual Life Plot or Parameter Stability Plot - see here.

Peaks-Over-Threshold (POT) method

Okay, if you have read previous post you should have a pretty good idea of what peaks-over-threshold (POT) method looks like now:

• We have an asymptotic result of convergence in distribution of some quantity, namely the exceedance over the threshold \(u\): \(X - u\), to a family of distribution parametrized by a pair of parameters.
• We don’t know these parameters and need to estimate them assuming the convergence takes place and for that there are the usual methods like MLE, bayesian inference, some papers also use Probability Weighted Moments.
• We can then verify the choice of \(u\) and the convergence assumption is reasonable a posteriori using qq plots or the stability plots previously mentioned.
• Once we are comfortable with our distribution we can use it to estimate whatever quantity make sense for us: VaR, Return values – the 100-year flood, Expected Shortfall… see this.

To wrap this section up, something that looks really interesting, that is introduced in the book from Coles and is still a bit far for me: the interepretation of these two theorem in term of limit of Poisson point process – that’s cool and that’s in chapter 7, will probably spend some time reading more about that in the future, you should too.

Next

Still no application and code! will need to address that shortly².

References

• An Introduction to Statistical Modeling of Extremes Values, Stuart Coles
• Pickands–Balkema–De Haan theorem - wikipedia
• An Application of Extreme Value Theory for Measuring Financial Risk, Manfred Gilli, et al