Extreme value theory

On the decay of awfulness with oftenness

In a satisfying way, it turns out that there are only so many shapes that probability densities can assume as they head off toward infinity. Extreme value theory makes this notion precise, and gives us some tools to work with them. Important application: understanding the kinds of heavy tailed variables we can observe in nature.

🏗


Tail limit theorems

The main result of use to our ends from EVT is the Pickands-Balkema-de Haan theorem .

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 , and is analysed in the EVT literature . 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.

🏗️

🏗️

References

Balkema, A. A., and L. de Haan. 1974. The Annals of Probability 2 (5): 792–804.
Castillo, Enrique, and Ali S. Hadi. 1997. Journal of the American Statistical Association 92 (440): 1609–20.
Charpentier, Arthur, and Emmanuel Flachaire. 2019. arXiv:1912.11736 [Econ, Stat], December.
Dargahi-Noubary, G. R. 1989. Mathematical Geology 21 (8): 829–42.
Davison, Anthony C. 1984. In Statistical Extremes and Applications, edited by J. Tiago de Oliveira, 461–82. NATO ASI Series. Dordrecht: Springer Netherlands.
Embrechts, Paul, S Kluppelberg, and Thomas Mikosch. 1997. Extremal Events in Finance and Insurance. Springer Berlin Heidelberg.
Embrechts, Paul, Claudia Klüppelberg, and Thomas Mikosch. 1997. Risk Theory.” In Modelling Extremal Events, 21–57. Applications of Mathematics 33. Springer Berlin Heidelberg.
Fisher, R. A., and L. H. C. Tippett. 1928. Mathematical Proceedings of the Cambridge Philosophical Society 24 (2): 180–90.
Ghitany, Mohamed, Emilio Gómez-Déniz, and Saralees Nadarajah. 2018. Journal of Risk and Financial Management 11 (1): 10.
Giesbrecht, F., and O. Kempthorne. 1976. Journal of the Royal Statistical Society: Series B (Methodological) 38 (3): 257–64.
Grimshaw, Scott D. 1993. Technometrics 35 (2): 185–91.
Hosking, J. R. M., and J. R. Wallis. 1987. Technometrics 29 (3): 339–49.
Hüsler, Jürg, Deyuan Li, and Mathias Raschke. 2011. Communications in Statistics - Theory and Methods 40 (14): 2500–2510.
Lee, Seyoon, and Joseph H. T. Kim. 2019. Communications in Statistics - Theory and Methods 48 (8): 2014–38.
Makarov, Mikhail. 2006. The Journal of Operational Risk 1 (2): 51–57.
Markovitch, Natalia M, and Udo R Krieger. 2002. Computer Networks 40 (3): 459–74.
McNeil, Alexander J. 1997. ASTIN Bulletin: The Journal of the IAA 27 (1): 117–37.
McNeil, Alexander J, Rüdiger Frey, and Paul Embrechts. 2005. Quantitative Risk Management : Concepts, Techniques and Tools. Princeton: Princeton Univ. Press.
Mueller, Ulrich K. 2018. arXiv:1802.00762 [Math], February.
Naveau, Philippe, Alexis Hannart, and Aurélien Ribes. 2020. Annual Review of Statistics and Its Application 7 (1): 89–110.
Nolde, Natalia, and Chen Zhou. 2021. Annual Review of Statistics and Its Application 8 (1): 217–40.
Pickands III, James. 1975. The Annals of Statistics 3 (1): 119–31.
Smith, Richard L. 1985. Biometrika 72 (1): 67–90.
Vajda, S. 1951. Scandinavian Actuarial Journal 1951 (1-2): 158–75.
Wong, T. S. T., and W. K. Li. 2006. In Institute of Mathematical Statistics Lecture Notes - Monograph Series, 272–83. Beachwood, Ohio, USA: Institute of Mathematical Statistics.
Zhao, Xu, Zhongxian Zhang, Weihu Cheng, and Pengyue Zhang. 2019. Mathematics 7 (5): 406.

No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.