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.

See also densities and intensities, survival analysis.

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\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

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

Balkema, A. A., and L. de Haan. 1974. “Residual Life Time at Great Age.” The Annals of Probability 2 (5, 5): 792–804. https://doi.org/10.1214/aop/1176996548.
Bhatti, Sajjad Haider, Shahzad Hussain, Tanvir Ahmad, Muhammad Aslam, Muhammad Aftab, and Muhammad Ali Raza. 2018. “Efficient Estimation of Pareto Model: Some Modified Percentile Estimators.” PLOS ONE 13 (5, 5): e0196456. https://doi.org/10.1371/journal.pone.0196456.
Castillo, Enrique, and Ali S. Hadi. 1997. “Fitting the Generalized Pareto Distribution to Data.” Journal of the American Statistical Association 92 (440, 440): 1609–20. https://doi.org/10.1080/01621459.1997.10473683.
Charpentier, Arthur, and Emmanuel Flachaire. 2019. “Pareto Models for Risk Management.” December 25, 2019. http://arxiv.org/abs/1912.11736.
Dargahi-Noubary, G. R. 1989. “On Tail Estimation: An Improved Method.” Mathematical Geology 21 (8, 8): 829–42. https://doi.org/10.1007/BF00894450.
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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. http://link.springer.com/chapter/10.1007/978-3-642-33483-2_2.
Fisher, R. A., and L. H. C. Tippett. 1928. “Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample.” Mathematical Proceedings of the Cambridge Philosophical Society 24 (2, 2): 180–90. https://doi.org/10.1017/S0305004100015681.
Ghitany, Mohamed, Emilio Gómez-Déniz, and Saralees Nadarajah. 2018. “A New Generalization of the Pareto Distribution and Its Application to Insurance Data.” Journal of Risk and Financial Management 11 (1, 1): 10. https://doi.org/10.3390/jrfm11010010.
Giesbrecht, F., and O. Kempthorne. 1976. “Maximum Likelihood Estimation in the Three-Parameter Lognormal Distribution.” Journal of the Royal Statistical Society: Series B (Methodological) 38 (3, 3): 257–64. https://doi.org/10.1111/j.2517-6161.1976.tb01591.x.
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Hosking, J. R. M., and J. R. Wallis. 1987. “Parameter and Quantile Estimation for the Generalized Pareto Distribution.” Technometrics 29 (3, 3): 339–49. https://doi.org/10.1080/00401706.1987.10488243.
Hüsler, Jürg, Deyuan Li, and Mathias Raschke. 2011. “Estimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit.” Communications in Statistics - Theory and Methods 40 (14, 14): 2500–2510. https://doi.org/10.1080/03610920903324874.
Lee, Seyoon, and Joseph H. T. Kim. 2019. “Exponentiated Generalized Pareto Distribution: Properties and Applications Towards Extreme Value Theory.” Communications in Statistics - Theory and Methods 48 (8): 2014–38. https://doi.org/10.1080/03610926.2018.1441418.
Makarov, Mikhail. 2006. “Extreme Value Theory and High Quantile Convergence.” The Journal of Operational Risk 1 (2, 2): 51–57. https://doi.org/10.21314/JOP.2006.009.
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