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.

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. β€œResidual Life Time at Great Age.” The Annals of Probability 2 (5): 792–804.
Bhatti, Sajjad Haider, Shahzad Hussain, Tanvir Ahmad, Muhammad Aslam, Muhammad Aftab, and Muhammad Ali Raza. 2018. β€œEfficient Estimation of Pareto Model: Some Modified Percentile Estimators.” PLOS ONE 13 (5): e0196456.
Castillo, Enrique, and Ali S. Hadi. 1997. β€œFitting the Generalized Pareto Distribution to Data.” Journal of the American Statistical Association 92 (440): 1609–20.
Charpentier, Arthur, and Emmanuel Flachaire. 2019. β€œPareto Models for Risk Management.” arXiv:1912.11736 [Econ, Stat], December.
Dargahi-Noubary, G. R. 1989. β€œOn Tail Estimation: An Improved Method.” Mathematical Geology 21 (8): 829–42.
Davison, Anthony C. 1984. β€œModelling Excesses over High Thresholds, with an Application.” 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 KlΓΌppelberg, and Thomas Mikosch. 1997. β€œRisk Theory.” In Modelling Extremal Events, 21–57. Applications of Mathematics 33. Springer Berlin Heidelberg.
Fisher, R. A., and L. H. C. Tippett. 1928. β€œLimiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample.” Mathematical Proceedings of the Cambridge Philosophical Society 24 (2): 180–90.
Ghitany, Mohamed, Emilio GΓ³mez-DΓ©niz, and Saralees Nadarajah. 2018. β€œA New Generalization of the Pareto Distribution and Its Application to Insurance Data.” Journal of Risk and Financial Management 11 (1): 10.
Giesbrecht, F., and O. Kempthorne. 1976. β€œMaximum Likelihood Estimation in the Three-Parameter Lognormal Distribution.” Journal of the Royal Statistical Society: Series B (Methodological) 38 (3): 257–64.
Grimshaw, Scott D. 1993. β€œComputing Maximum Likelihood Estimates for the Generalized Pareto Distribution.” Technometrics 35 (2): 185–91.
Hosking, J. R. M., and J. R. Wallis. 1987. β€œParameter and Quantile Estimation for the Generalized Pareto Distribution.” Technometrics 29 (3): 339–49.
HΓΌsler, JΓΌrg, Deyuan Li, and Mathias Raschke. 2011. β€œEstimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit.” Communications in Statistics - Theory and Methods 40 (14): 2500–2510.
Lee, Seyoon, and Joseph H. T. Kim. 2019. β€œExponentiated Generalized Pareto Distribution: Properties and Applications Towards Extreme Value Theory.” Communications in Statistics - Theory and Methods 48 (8): 2014–38.
Makarov, Mikhail. 2006. β€œExtreme Value Theory and High Quantile Convergence.” The Journal of Operational Risk 1 (2): 51–57.
Markovitch, Natalia M, and Udo R Krieger. 2002. β€œThe Estimation of Heavy-Tailed Probability Density Functions, Their Mixtures and Quantiles.” Computer Networks 40 (3): 459–74.
McNeil, Alexander J. 1997. β€œEstimating the Tails of Loss Severity Distributions Using Extreme Value Theory.” ASTIN Bulletin: The Journal of the IAA 27 (1): 117–37.
McNeil, Alexander J, RΓΌdiger Frey, and Paul Embrechts. 2005. Quantitative Risk Management : Concepts, Techniques and Tools. Princeton: Princeton Univ. Press.
Mueller, Ulrich K. 2018. β€œRefining the Central Limit Theorem Approximation via Extreme Value Theory.” arXiv:1802.00762 [Math], February.
Naveau, Philippe, Alexis Hannart, and AurΓ©lien Ribes. 2020. β€œStatistical Methods for Extreme Event Attribution in Climate Science.” Annual Review of Statistics and Its Application 7 (1): 89–110.
Nolde, Natalia, and Chen Zhou. 2021. β€œExtreme Value Analysis for Financial Risk Management.” Annual Review of Statistics and Its Application 8 (1): 217–40.
Pickands III, James. 1975. β€œStatistical Inference Using Extreme Order Statistics.” The Annals of Statistics 3 (1): 119–31.
Smith, Richard L. 1985. β€œMaximum Likelihood Estimation in a Class of Nonregular Cases.” Biometrika 72 (1): 67–90.
Vajda, S. 1951. β€œAnalytical Studies in Stop-Loss Reinsurance.” Scandinavian Actuarial Journal 1951 (1-2): 158–75.
Wong, T. S. T., and W. K. Li. 2006. β€œA Note on the Estimation of Extreme Value Distributions Using Maximum Product of Spacings.” 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. β€œA New Parameter Estimator for the Generalized Pareto Distribution Under the Peaks over Threshold Framework.” Mathematics 7 (5): 406.

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